This page contains a complete infinite series exam with worked out solutions.
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IMPORTANT 
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Exam Details  

Time  1 hour 
Questions  13 
Total Points  100 
Tools  

Calculator  not allowed 
Formula Sheet(s) 

Other Tools  none 
Instructions: 

 Show all your work. 
Section 1
3 questions  20 points total 

Do the following sequences \( \{ {a_n} \} \) converge or diverge as \( n \to \infty \)? If the sequence converges, find it's limit. Justify your answers.
Does the sequences \( \{ {a_n} \} \) where \(a_n=2+(1)^n\) converge or diverge as \( n \to \infty \)? If the sequence converges, find it's limit. Justify your answers.
Problem Statement 

Does the sequences \( \{ {a_n} \} \) where \(a_n=2+(1)^n\) converge or diverge as \( n \to \infty \)? If the sequence converges, find it's limit. Justify your answers.
Final Answer 

The sequence \( \left\{ 2 + (1)^n \right\}\) diverges.
Problem Statement 

Does the sequences \( \{ {a_n} \} \) where \(a_n=2+(1)^n\) converge or diverge as \( n \to \infty \)? If the sequence converges, find it's limit. Justify your answers.
Solution 

The sequence diverges since it oscillates between 1 and 3.
For example, if \(\epsilon=1\), there is no number \(L\) such that \(  a_nL  < \epsilon\) for all sufficiently large \(n\), since then we would have both \(  1L  < 1\) (or \(0 < L < 2\) ) and \(  3L  < 1\) or \((2 < L < 4)\), which is impossible. So there is no \(L\) that satisfies the definition of the limit.
Final Answer 

The sequence \( \left\{ 2 + (1)^n \right\}\) diverges. 
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Does the sequence \( \{ {a_n} \} \) where \(\displaystyle{ a_n = \frac{n}{e^n} }\) converge or diverge as \( n \to \infty \)? If the sequence converges, find it's limit. Justify your answer.
Problem Statement 

Does the sequence \( \{ {a_n} \} \) where \(\displaystyle{ a_n = \frac{n}{e^n} }\) converge or diverge as \( n \to \infty \)? If the sequence converges, find it's limit. Justify your answer.
Final Answer 

The sequence \(\displaystyle{ \left\{ \frac{n}{e^n} \right\} }\) converges to zero.
Problem Statement 

Does the sequence \( \{ {a_n} \} \) where \(\displaystyle{ a_n = \frac{n}{e^n} }\) converge or diverge as \( n \to \infty \)? If the sequence converges, find it's limit. Justify your answer.
Solution 

Looking at the limit \(\displaystyle{ \lim_{n \to \infty}{\frac{n}{e^n}} }\), direct substitution yields \( \infty / \infty \) which is indeterminate. So we can use L'HÃ´pital's Rule.
\(\displaystyle{ \lim_{n \to \infty}{\frac{n}{e^n}} = \lim_{n \to \infty}{\frac{1}{e^n}} = \frac{1}{\infty} = 0 }\)
Therefore, the sequence converges to zero.
Final Answer 

The sequence \(\displaystyle{ \left\{ \frac{n}{e^n} \right\} }\) converges to zero. 
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Does the sequence \( \{ {a_n} \} \) where \(\displaystyle{ a_n = \left( 1 + \frac{2}{n} \right)^n }\) converge or diverge as \( n \to \infty \)? If the sequence converges, find it's limit. Justify your answers.
Problem Statement 

Does the sequence \( \{ {a_n} \} \) where \(\displaystyle{ a_n = \left( 1 + \frac{2}{n} \right)^n }\) converge or diverge as \( n \to \infty \)? If the sequence converges, find it's limit. Justify your answers.
Final Answer 

The sequence \(\displaystyle{ \left\{ \left( 1 + \frac{2}{n} \right)^n \right\} }\) converges to \( e^2 \).
Problem Statement 

Does the sequence \( \{ {a_n} \} \) where \(\displaystyle{ a_n = \left( 1 + \frac{2}{n} \right)^n }\) converge or diverge as \( n \to \infty \)? If the sequence converges, find it's limit. Justify your answers.
Solution 

Direct substitution yields \( \infty ^ {\infty} \) which is indeterminate. We don't have a fraction, so we can't use L'HÃ´pital's Rule without doing some algebra.
\( y = \displaystyle{ \lim_{n \to \infty}{ \left(1 + \frac{2}{n}\right)^n } }\) 
Take the natural log of both sides. 
\( \ln(y) = \displaystyle{ \lim_{n \to \infty}{\ln \left(1 + \frac{2}{n}\right)^n }} \) 
\( \ln(y) = \displaystyle{ \lim_{n \to \infty}{(n) \ln \left(1 + \frac{2}{n}\right) }}\) 
\( \ln(y) = \displaystyle{\lim_{n \to \infty}{ \frac{\ln \left(1 + \frac{2}{n}\right)}{1/n} }} \) 
\( \ln(y) = \displaystyle{\lim_{n \to \infty}{ \frac{\ln [(n+2)/n]}{n^{1}} }}\) 
Use L'HÃ´pital's Rule. 
\( \ln(y) = \displaystyle{\lim_{n \to \infty}{ \left[ \frac{2n^{2}(n)}{n+2} \cdot \frac{1}{n^{2}} \right] }} \) 
\( \ln(y) = \displaystyle{\lim_{n \to \infty}{ \frac{2n}{n+2} }}\) 
Use L'HÃ´pital's Rule again. 
\( \ln(y) = \displaystyle{\lim_{n \to \infty}{\frac{2}{1}}}\) 
\( \ln(y) = 2 \) 
Undo the natural log by raising each side to the exponent of e. 
\( e^{\ln(y)} = e^2 \) 
\( y = e^2 \) 
Final Answer 

The sequence \(\displaystyle{ \left\{ \left( 1 + \frac{2}{n} \right)^n \right\} }\) converges to \( e^2 \). 
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Section 2
4 questions  20 points total 

Determine whether the following series converge or diverge. You may use any appropriate test provided you explain your answer.
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3}{3^n} } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3}{3^n} } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Final Answer 

The series \(\displaystyle{ \sum_{n=1}^{\infty}{\frac{n^3}{3^n}} }\) converges by the Ratio Test.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3}{3^n} } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Solution 

Let's go down through the infinite series table in order.
Let \(\displaystyle{ a_n = \frac{n^3}{3^n} }\)
Group 1: Divergence Test: Using L'HÃ´pital's Rule repeatedly, we can eventually get to the point where we can see that the limit \( \displaystyle{ \lim_{n \to \infty}{a_n} = 0 }\), so the test is inconclusive.
Group 2: We don't have one of the special series, pSeries, Geometric Series, Alternating Series or Telescoping Series.
Group 3: So, let's try the Ratio Test.
We need to set up the limit \(\displaystyle{ \lim_{n \to \infty}{\frac{a_{n+1}}{a_n}} }\).
Since \(\displaystyle{ a_n = \frac{n^3}{3^n} }\), \(\displaystyle{ a_{n+1} = \frac{(n+1)^3}{3^{n+1}} }\).
\( \displaystyle{\lim_{n \to \infty}{\frac{a_{n+1}}{a_n}}} \) 
\( \displaystyle{\lim_{n \to \infty}{ \frac{(n+1)^3}{3^{n+1}} \cdot \frac{3^n}{n^3} }} \) 
\( \displaystyle{\lim_{n \to \infty}{ \frac{(n+1)^3}{n^3} \cdot \frac{3^n}{3^{n+1}} }} \) 
\( \displaystyle{\lim_{n \to \infty}{ \left( \frac{n+1}{n} \right)^3 \cdot \frac{1}{3} }} \) 
\( \displaystyle{\frac{1}{3} \lim_{n \to \infty}{ \left( 1 + \frac{1}{n} \right)^3 }} \) 
\( \displaystyle{\frac{1}{3} (1 + 0)^3 = \frac{1}{3}} \) 
Since \(\displaystyle{ \lim_{n \to \infty}{\frac{a_{n+1}}{a_n}} = 1/3 < 1 }\) the series converges.
Final Answer 

The series \(\displaystyle{ \sum_{n=1}^{\infty}{\frac{n^3}{3^n}} }\) converges by the Ratio Test. 
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Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{\frac{n}{n^2+1} } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{\frac{n}{n^2+1} } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Final Answer 

The series \(\displaystyle{ \sum_{n=1}^{\infty}{\frac{n}{n^2+1} } }\) diverges by the Limit Comparison Test.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{\frac{n}{n^2+1} } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Solution 

Let's go down through the infinite series tests in the order they appear in the infinite series table.
Group 1: Divergence Test: Using L'HÃ´pital's Rule, the limit goes to zero, so the test is inconclusive.
Group 2: This series is not one of the special series.
Group 3: Start with the Ratio Test. Looking at the series, our coefficients and constants are all one, so we might think that the limit would equal one, which is correct.
Next, we can try a comparison test. But what do we compare this series to? Well, as n gets very large, the \(1\) in the denominator becomes almost negligible compared to the \(n^2\). So let's compare this series to \(\displaystyle{ \sum{\frac{n}{n^2}} = \sum{\frac{1}{n}} }\), which we know is a pSeries with \(p=1\), so it diverges.
We can use either the Limit Comparison Test or the Direct Comparison Test. Let's try the Limit Comparison Test.
We need to set up the limit \(\displaystyle{ \lim_{n \to \infty}{\frac{a_n}{t_n}} }\) where \(\displaystyle{ a_n = \frac{n}{n^2+1} }\) and \(\displaystyle{ t_n = \frac{1}{n} }\) which is our divergent test series. If this limit is finite and positive then the original series will also diverge. Let's see what happens.
\(\begin{array}{rcl} \displaystyle{\lim_{n \to \infty}{\frac{a_n}{t_n}}} & = & \displaystyle{\lim_{n \to \infty}{ \frac{n}{n^2+1} \cdot \frac{n}{1} }} \\\\ & = & \displaystyle{\lim_{n \to \infty}{ \frac{n^2}{n^2 + 1} }} \\\\ & = & \displaystyle{\lim_{n \to \infty}{ \frac{2n}{2n} } = 1} \end{array}\)
In the second step, we used L'HÃ´pital's Rule.
Since the limit is finite and positive and the test series diverges, the original series also diverges.
Final Answer 

The series \(\displaystyle{ \sum_{n=1}^{\infty}{\frac{n}{n^2+1} } }\) diverges by the Limit Comparison Test. 
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Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{n \sin\left( \frac{1}{n} \right) } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{n \sin\left( \frac{1}{n} \right) } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Final Answer 

The series \(\displaystyle{ \sum_{n=1}^{\infty}{n \cdot \sin\left( \frac{1}{n} \right) } }\) diverges by the Divergence Test.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{n \sin\left( \frac{1}{n} \right) } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Solution 

There are several ways to work this one. The most straightforward is to go through the infinite series table in order.
Group 1  Divergence Test: Evaluate the limit \(\displaystyle{ \lim_{n \to \infty}{ n \cdot \sin\left( \frac{1}{n} \right) } }\). Direct substitution gives us \( \infty \cdot 0 \) which is indeterminate. So we can use L'HÃ´pital's Rule if we get the limit in a fraction.
\( \displaystyle{\lim_{n \to \infty}{ n \cdot \sin\left( \frac{1}{n} \right) } }\) 
\( \displaystyle{\lim_{n \to \infty}{ \frac{\sin(1/n)}{1/n} }} \) 
\( \displaystyle{\lim_{n \to \infty}{ \frac{\sin(n^{1})}{n^{1}} }} \) 
\( \displaystyle{\lim_{n \to \infty}{ \frac{\cos(n^{1}) (1)(n^{2})}{(1)n^{2}} }} \) 
\( \displaystyle{\lim_{n \to \infty}{ \cos(n^{1}) } = \cos(0) = 1} \) 
So, since the limit does not equal zero, the series diverges.
Note: Another way to evaluate this limit is to substitute \( x = 1/n \) and as \( n \to \infty \), \( x \to 0 \). So the limit is \(\displaystyle{ \lim_{n \to \infty}{ n \cdot \sin\left( \frac{1}{n} \right) } = \lim_{n \to \infty}{ \frac{\sin\left( \frac{1}{n} \right)}{1/n} } = \lim_{x \to 0}{\frac{\sin(x)}{x}} }\)
The last limit is a special trig limit which is equal to one. If you didn't remember this, you could have used L'HÃ´pital's Rule again here and arrived at the same answer.
Final Answer 

The series \(\displaystyle{ \sum_{n=1}^{\infty}{n \cdot \sin\left( \frac{1}{n} \right) } }\) diverges by the Divergence Test. 
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Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left( \frac{1}{\sqrt{n}}  \frac{1}{\sqrt{n+1}}\right) } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left( \frac{1}{\sqrt{n}}  \frac{1}{\sqrt{n+1}}\right) } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Final Answer 

The Telescoping Series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left( \frac{1}{\sqrt{n}}  \frac{1}{\sqrt{n+1}}\right) } }\) converges.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left( \frac{1}{\sqrt{n}}  \frac{1}{\sqrt{n+1}}\right) } }\) converges or diverges. You may use any appropriate test provided you explain your answer.
Solution 

Without too much work, this series is crying out Telescoping Series. Let's set up a table to see if we can find a pattern.
\(\begin{array}{ccc} n & & a_n \\ 1 & & \displaystyle{\frac{1}{1}  \frac{1}{\sqrt{2}}} \\ 2 & & \displaystyle{\frac{1}{\sqrt{2}}  \frac{1}{\sqrt{3}}} \\ 3 & & \displaystyle{\frac{1}{\sqrt{3}}  \frac{1}{\sqrt{4}}} \\ 4 & & \displaystyle{\frac{1}{\sqrt{4}}  \frac{1}{\sqrt{5}}} \end{array}\)
With only 4 terms, we can see that the last term of each line cancels with the first term in the next line.
This gives us \( \displaystyle{ 1  \lim_{n \to \infty}{\frac{1}{\sqrt{n+1}}} = 1 }\)
So the series converges (specifically to 1).
Final Answer 

The Telescoping Series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left( \frac{1}{\sqrt{n}}  \frac{1}{\sqrt{n+1}}\right) } }\) converges. 
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Section 3
2 questions  20 points total 

Determine whether the following series converge or diverge and justify your answer.
Determine whether the series \(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n \ln(n)} } }\) converges or diverges and justify your answer.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n \ln(n)} } }\) converges or diverges and justify your answer.
Final Answer 

The series \(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n~\ln(n)} } }\) diverges by the Integral Test.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n \ln(n)} } }\) converges or diverges and justify your answer.
Solution 

Going through the infinite series table in order will help us determine what rule to use here.
Group 1: Divergence Test: The limit here is obviously zero, so this test is inconclusive.
Group 2: This series is not one of the special series.
Group 3: From experience you probably know that, using the Ratio Test or a comparison test, would yield a very difficult limit or inequality to evaluate due to the \( \ln(n) \) term. So, let's try the Integral Test.
We have \(\displaystyle{ f(n) = \frac{1}{n~\ln(n)} }\) which is a discrete function. To use the integral test, we need a continuous function. So let \(\displaystyle{ f(x) = \frac{1}{x~\ln(x)} }\).
The function \( f(x) \) is continuous and positive on \( 2 \leq x < \infty \). It is also decreasing on this interval since the denominator is increasing and the numerator is constant. Therefore, by the integral test, the series converges or diverges with the integral \(\displaystyle{ \int_{2}^{\infty}{\frac{1}{x~\ln(x)} dx } }\). \(\displaystyle{ \int_{2}^{\infty}{\frac{1}{x~\ln(x)} dx } = \lim_{b \to \infty}{ \int_{2}^{b}{\frac{1}{x~\ln(x)} dx} }}\)
Using the technique of substitution, we let \( u = \ln(x) \to du = dx/x \).
\(\displaystyle{ \int{\frac{1}{x~\ln(x)} dx} = }\) \(\displaystyle{ \int{\frac{1}{u} du} = \ln(u) = \ln( \ln(x) ) }\)
\(\displaystyle{\lim_{b \to \infty}{ \int_{2}^{b}{\frac{1}{x~\ln(x)} dx} } = }\) \(\displaystyle{\lim_{b \to \infty}{ \left[ \ln( \ln(x) ) \right]_2^b } = }\) \(\displaystyle{\lim_{b \to \infty}{ [ \ln(\ln(b))  \ln(\ln(2)) ] } = }\) \(\displaystyle{ \infty  \ln(\ln(2)) = \infty }\)
Since the integral diverges, the series also diverges.
Final Answer 

The series \(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n~\ln(n)} } }\) diverges by the Integral Test. 
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Determine whether the series \(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{(1)^n \ln(n)}{n} } }\) converges or diverges and justify your answer.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{(1)^n \ln(n)}{n} } }\) converges or diverges and justify your answer.
Final Answer 

The series \(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{(1)^n \ln(n)}{n} } }\) converges by the Alternating Series Test.
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{(1)^n \ln(n)}{n} } }\) converges or diverges and justify your answer.
Solution 

We don't really need to go through the tests in the infinite series table since we can see that this is an alternating series. So, let's try the Alternating Series Test first.
For convergence, we need to verify that both of the following conditions hold.
Condition 1: \(\displaystyle{ \lim_{n \to \infty}{a_n} = 0 }\)
Condition 2: \( 0 < a_{n+1} < a_n \)
Let's test condition 1.
\(\displaystyle{\lim_{n \to \infty}{a_n} = \lim_{n \to \infty}{\frac{\ln(x)}{x}} = \lim_{n \to \infty}{\frac{1/x}{1}} = 0 }\)
We used L'HÃ´pital's Rule to show that condition 1 holds. Now, let's test condition 2.
\(\displaystyle{ a_n = \frac{\ln(n)}{n} \text{ and } a_{n+1} = \frac{\ln(n+1)}{n+1} }\)
\(\displaystyle{ 0 < a_{n+1} < a_n \to 0 < \frac{\ln(n+1)}{n+1} < \frac{\ln(n)}{n} } \)
Since \( \ln(n) \) and \( n \) are both positive, we can drop the \( 0 < \) part of the inequality.
However, it is difficult to establish the inequality \(\displaystyle{ \frac{\ln(n+1)}{n+1} < \frac{\ln(n)}{n} }\)
So, we need to use another technique. The idea of the inequality is that the terms are decreasing. We can set up a continuous function \(\displaystyle{ f(x) = \frac{\ln(x)}{x} }\) and show that it is decreasing, i.e. the slope is negative. Let's calculate the derivative and see if it works.
\(\displaystyle{ f'(x) = \frac{x \cdot 1/x  1 (\ln(x))}{x^2} = \frac{1  \ln(x)}{x^2} }\)
We used the Quotient Rule here.
Looking at the derivative, the denominator \( x^2 \) is always positive, so the numerator will determine the sign. We want the slope to be negative, so \( 1  \ln(x) < 0 \) or \( 1 < \ln(x) \). This inequality holds for \( x > e \). So as long as \( n \geq 3 \), the terms are decreasing. Therefore, condition 2 holds and so the alternating series converges.
Final Answer 

The series \(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{(1)^n \ln(n)}{n} } }\) converges by the Alternating Series Test. 
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Section 4
4 questions  40 points total 

Does this series diverge, converge conditionally or converge absolutely? Justify your answer.
\[ A = 1  \frac{1}{2^2} + \frac{1}{3^2}  \frac{1}{4^2} + \frac{1}{5^2}  \frac{1}{6^2} + \frac{1}{7^2}  . . . \]
Problem Statement 

Does this series diverge, converge conditionally or converge absolutely? Justify your answer.
\[ A = 1  \frac{1}{2^2} + \frac{1}{3^2}  \frac{1}{4^2} + \frac{1}{5^2}  \frac{1}{6^2} + \frac{1}{7^2}  . . . \]
Final Answer 

The series converges absolutely.
Problem Statement 

Does this series diverge, converge conditionally or converge absolutely? Justify your answer.
\[ A = 1  \frac{1}{2^2} + \frac{1}{3^2}  \frac{1}{4^2} + \frac{1}{5^2}  \frac{1}{6^2} + \frac{1}{7^2}  . . . \]
Solution 

First let's get a closed form for the series. Since this is an alternating series, we need a term that looks like \( (1)^n \). And looking at the denominator, it looks like \( n^2 \). We also need to determine where to start \(n\), \( 0 \) or \( 1 \). If we start with zero, we will get a zero in the denominator, which is not good. So we want to start at \( n = 1 \). Since the first term is positive, the power on the \( 1 \) term needs to be \( n + 1 \). So our series is \(\displaystyle{ \sum_{n=1}^{\infty}{\frac{(1)^{n+1}}{n^2}} }\).
We know that the absolute value of the alternating series, \(\displaystyle{ \sum{\frac{1}{n^2}} }\), is a convergent pSeries. So the Alternating Series converges absolutely.
Final Answer 

The series converges absolutely. 
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Let \(\displaystyle{S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + }\) \(\displaystyle{ \frac{1}{4^2} + \frac{1}{5^2} + }\) \(\displaystyle{\frac{1}{6^2} + \frac{1}{7^2} + . . . }\)
Given that \( S = \pi^2/6 \), find the sum A given in the previous question.
(Hint: Consider \( S  A \) and express it in terms of \( S \).)
Problem Statement 

Let \(\displaystyle{S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + }\) \(\displaystyle{ \frac{1}{4^2} + \frac{1}{5^2} + }\) \(\displaystyle{\frac{1}{6^2} + \frac{1}{7^2} + . . . }\)
Given that \( S = \pi^2/6 \), find the sum A given in the previous question.
(Hint: Consider \( S  A \) and express it in terms of \( S \).)
Final Answer 

\(\displaystyle{ A = \frac{\pi^2}{12} }\)
Problem Statement 

Let \(\displaystyle{S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + }\) \(\displaystyle{ \frac{1}{4^2} + \frac{1}{5^2} + }\) \(\displaystyle{\frac{1}{6^2} + \frac{1}{7^2} + . . . }\)
Given that \( S = \pi^2/6 \), find the sum A given in the previous question.
(Hint: Consider \( S  A \) and express it in terms of \( S \).)
Solution 

By the linearity of limits and sums, if the series \( \sum{a_n} \) and \( \sum{b_n} \) converge, then \( \sum{(a_n  b_n) } \) also converges and \( \sum{(a_n  b_n) } = \sum{a_n}  \sum{b_n} \). Also, for any constant \(c\), we have \( \sum{ca_n} = c \sum{a_n} \).
Since the series for \(S\) and \(A\) both converge, so we can form \(SA\).
Follow the hint and form \(SA\). 
\(\displaystyle{\left( 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + . . . \right)  }\) \(\displaystyle{ \left( 1  \frac{1}{2^2} + \frac{1}{3^2}  \frac{1}{4^2} + \frac{1}{5^2}  . . . \right)}\) 
Pair up terms that match to see what cancels. 
\(\displaystyle{(1  1) + \left( \frac{1}{2^2} + \frac{1}{2^2} \right) + \left( \frac{1}{3^2}  \frac{1}{3^2} \right) + }\) \(\displaystyle{ \left( \frac{1}{4^2} + \frac{1}{4^2} \right) + \left( \frac{1}{5^2}  \frac{1}{5^2} \right) + . . . }\) 
\(\displaystyle{2 \left( \frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + \frac{1}{8^2} + . . . \right)}\) 
Factor 
\(\displaystyle{\frac{2}{2^2} \left( 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + . . . \right)}\) 
Notice that the term on the right is \(S\). 
\(\displaystyle{\frac{1}{2}S}\) 
So now we have \( S  A = S/2 \) and solving for \(A\), we get \( A = S/2 \). Since \( S = \pi^2/6 \), it follows that \( A = \pi^2 / 12 \).
Final Answer 

\(\displaystyle{ A = \frac{\pi^2}{12} }\) 
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State the definition for a sequence \( \{ a_n \} \) to converge to a limit \( L \) as \( n \to \infty \).
Problem Statement 

State the definition for a sequence \( \{ a_n \} \) to converge to a limit \( L \) as \( n \to \infty \).
Solution 

A sequence \( \{ a_n \} \) converges to a limit \( L \) if, for every \( \epsilon > 0 \) there exists a number \(N\) such that \[  a_n  L  < \epsilon ~~~~~ \text{ for every } n > N \]
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If \(\displaystyle{ a_n = \frac{1}{\sqrt{n}} }\) for \( n = 1, 2, 3, . . .\) prove from the definition that \(\displaystyle{ \lim_{n \to \infty}{a_n} = 0 }\).
Problem Statement 

If \(\displaystyle{ a_n = \frac{1}{\sqrt{n}} }\) for \( n = 1, 2, 3, . . .\) prove from the definition that \(\displaystyle{ \lim_{n \to \infty}{a_n} = 0 }\).
Solution 

Given \( \epsilon > 0 \), let \( N = 1/\epsilon^2 \). Then if \( n > N \), we have
\(\begin{array}{rcl} \displaystyle{\left \frac{1}{\sqrt{n}}  0 \right} & = & \displaystyle{\frac{1}{\sqrt{n}}} \\ & < & \displaystyle{\frac{1}{\sqrt{N}}} \\ & < & \epsilon \end{array}\)
Since \( 1/\sqrt{N} = \epsilon \), it follows that \(\displaystyle{ \lim_{n \to \infty}{\frac{1}{\sqrt{n}}} = 0 }\).
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