17Calculus Infinite Series - p-Series
17Calculus
p-Series Convergence Theorem | ||
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\(\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n^p}} = 1 + \frac{1}{2^p} + \frac{1}{3^p} + \frac{1}{4^p} + ~~ . . . }\) | ||
\( 0 \lt p \leq 1\) |
\( p>1\) | |
diverges |
converges | |
Note - In the proof section below, we show a more efficient way to write this theorem statement.
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The p-series is a pretty straight-forward series to understand and use. One detail you need to notice is that the theorem covers cases only where \( p > 0\).
Question: Do you know what happens when \( p \leq 0\)? |
Okay, so testing for convergence and divergence of p-series looks pretty easy and, fortunately, it is. But it is also very powerful. To help you cement the idea more firmly in your mind, read through the panel below discussing the proof of this test. You do not have to understand the integral test at this point to understand this proof.
p-Series Convergence Theorem Proof
p-Series Convergence Theorem |
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The p-series \(\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n^p}}}\)
diverges |
converges | |
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\( 0 \lt p \leq 1\) |
\( p>1\) |
To prove this, we will use the Integral Test and the Special Improper Integral.
The integral test tells us that we can set up the integral \(\displaystyle{ \int_1^{\infty}{\frac{1}{x^p} dx} }\)
If this integral converges, then so does the series. Similarly, if the integral diverges, the series also diverges.
We will look at these five cases.
1. \( p > 1 \)
2. \( p = 1 \)
3. \( 0 \lt p \leq 1\)
4. \( p = 0 \)
5. \( p \lt 0 \)
Note: Although the last two cases are not part of the theorem, we will show what happens in those two cases to answer the question at the top of the page and for completeness.
Let's use the Integral Test to prove convergence and divergence by calculating the corresponding improper integrals. For most of the cases, we need to set up a limit as follows
\(\displaystyle{ \lim_{b \to \infty}{ \int_1^{b}{\frac{1}{x^p} dx}} }\)
Since the function \(\displaystyle{ \frac{1}{x^p} }\) is continuous on the interval \( [1,b] \), this integral can be evaluated. So, now let's look at each case individually.
Case 1: \( p > 1 \)
When \( p > 1 \), the integral is continuous and decreasing on the interval. So the Integral Test applies.
\(\displaystyle{ \lim_{b \to \infty}{ \int_1^{b}{\frac{1}{x^p} dx}} }\) |
\(\displaystyle{ \lim_{b \to \infty}{ \int_1^{b}{ x^{-p} dx}} }\) |
\(\displaystyle{ \lim_{b \to \infty}{ \left[ \frac{x^{-p+1}}{-p+1} \right]_{1}^{b} } }\) |
\(\displaystyle{ \frac{1}{-p+1} \lim_{b \to \infty}{\left[ b^{-p+1} - 1^{-p+1} \right]} }\) |
\(\displaystyle{ \frac{1}{-p+1} [ 0 - 1 ] = \frac{1}{p-1} }\) |
Since \(\displaystyle{ \frac{1}{p-1} }\) is finite, the integral converges and, therefore, by the Integral Test, the series also converges.
Case 2: \( p=1 \)
When \(p = 1\), we have
\(\displaystyle{
\lim_{b \to \infty}{ \int_1^{b}{\frac{1}{x} dx}} \to \lim_{b \to \infty }{ \left[ \ln(x) \right]_{1}^{b} } \to
\lim_{b \to \infty }{ \ln(b) } - 0 = \infty
}\)
Since the limit is infinity, the series diverges.
Case 3: \( 0 \lt p \lt 1\)
We will use the Integral Test again.
\(\displaystyle{ \lim_{b \to \infty}{ \int_1^{b}{\frac{1}{x^p} dx}} }\) |
\(\displaystyle{ \lim_{b \to \infty}{ \int_1^{b}{ x^{-p} dx}} }\) |
\(\displaystyle{ \lim_{b \to \infty}{ \left[ \frac{x^{-p+1}}{-p+1} \right]_{1}^{b} } }\) |
\(\displaystyle{ \frac{1}{-p+1} \lim_{b \to \infty}{\left[ b^{-p+1} - 1^{-p+1} \right]} }\) |
So far in these calculations, we have the same equation as we did in case 1. However, in this case, \( 0 \lt p \lt 1\), which means that the exponent \(-p+1 > 0\) and therefore \(\displaystyle{ \lim_{b \to \infty}{ b^{-p+1} } \to \infty }\). So the entire integral diverges, which means that the series diverges.
Case 4: \( p = 0 \)
When \( p = 0 \), the series is \(\displaystyle{ \sum_{n=1}^{\infty}{1} }\). Since \(\displaystyle{ \lim_{n \to \infty}{1} = 1 \neq 0 }\) the divergence test tells us that the series diverges.
Case 5: \( p \lt 0 \)
We can write the fraction \(\displaystyle{ \frac{1}{n^p} = n^{-p} }\). Since \( p \lt 0 \), the exponent here is positive and so the terms are increasing. Since \(\displaystyle{ \lim_{n \to \infty}{ \frac{1}{n^p} } \neq 0 }\), the series diverges by the divergence test.
In Summary: For the series \(\displaystyle{\sum_{n=1}^{\infty}{\frac{1}{n^p}}}\)
1. \( p > 1 \) converges by the integral test
2. \( p = 1 \) diverges by the integral test
3. \( 0 \lt p \lt 1\) diverges by the integral test
4. \( p = 0 \) diverges by the divergence test
5. \( p \lt 0 \) diverges by the divergence test
- qed -
Note: We have also seen the p-series theorem where, instead of writing \( 0 \lt p \leq 1\) for where the series diverges, it says the series diverges for \( p \leq 1 \). Of course, this includes all of the cases above. This is a more efficient way to write the theorem statement.
Euler's Constant
Here is a great video discussing one use of a p-series.
Note - - Euler's Constant is not the same as Euler's Number.
PatrickJMT - Euler's Constant [16min-10secs]
video by PatrickJMT |
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Practice
Unless otherwise instructed, determine whether these series converge or diverge using the p-series convergence theorem.
\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n} } }\)
Problem Statement |
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Determine the convergence or divergence of the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n} } }\) using the Direct Comparison Test.
Final Answer |
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The series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n} } }\) diverges by the Direct Comparison Test.
Problem Statement
Determine the convergence or divergence of the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n} } }\) using the Direct Comparison Test.
Solution
There are at least two ways to think about this one.
p-Series
First, if you factor out the constant in the numerator, you get \(\displaystyle{ 2 \sum_{n=1}^{\infty}{ \frac{1}{n} } }\) and the remaining sum is just a p-series with \(p=1\). Therefore, the series diverges. However, the problem statement requires us to use the DCT.
Direct Comparison Test
Using the Direct Comparison Test, we compare the given series with \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n} } }\) which is a divergent p-series.
Now, to confirm that the given series diverges, we need to test \(\displaystyle{ \frac{2}{n} \geq \frac{1}{n} }\) for all \(n\). If we multiply both sides of the inequality by \(n\) (which we can do since \(n\) is always positive), we get \( 2 \geq 1 \). This is true for all \(n\), therefore the series diverges.
Final Answer
The series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n} } }\) diverges by the Direct Comparison Test.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }\)
Problem Statement |
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Determine the convergence or divergence of the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }\) using the integral test, if possible.
Final Answer |
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The series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }\) converges by the p-series test or the integral test.
Problem Statement
Determine the convergence or divergence of the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }\) using the integral test, if possible.
Solution
This is a p-series with \( p=2 > 1\), so the series converges by the p-series test.
We could also have used the Integral Test, as follows.
\(\displaystyle{ \int_{1}^{\infty}{\frac{1}{x^2}dx} }\) |
\(\displaystyle{ \lim_{b \to \infty}{\int_{1}^{b}{\frac{1}{x^2}dx}} }\) |
\(\displaystyle{ \lim_{b \to \infty}{\int_{1}^{b}{x^{-2}dx}} }\) |
\(\displaystyle{ \lim_{b \to \infty}{ \left[ -x^{-1} \right]_{1}^{b}} }\) |
\(\displaystyle{ \lim_{b \to \infty}{-b^{-1} + 1^{-1}} }\) |
\( 0 + 1 = 1 \) |
Since the improper integral is finite, the series converges by the Integral Test.
Note: The value \(1\), from the integral is NOT necessarily what the series converges to. The significance of this number is only that it is finite.
Final Answer
The series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^2} } }\) converges by the p-series test or the integral test.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n^2} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n^2} } }\) converges or diverges using the p-series convergence theorem.
Solution
3163 video solution
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1.4}} } }\)
Problem Statement |
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Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1.4}} } }\) converges or diverges using the p-series convergence theorem.
Final Answer |
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The series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1.4}} } }\) is a convergent p-series.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1.4}} } }\) converges or diverges using the p-series convergence theorem.
Solution
PatrickJMT - 253 video solution
video by PatrickJMT |
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Final Answer
The series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1.4}} } }\) is a convergent p-series.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{0.7}} } }\)
Problem Statement |
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Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{0.7}} } }\) converges or diverges using the p-series convergence theorem.
Final Answer |
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The series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{0.7}} } }\) is a divergent p-series.
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{0.7}} } }\) converges or diverges using the p-series convergence theorem.
Solution
PatrickJMT - 254 video solution
video by PatrickJMT |
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Final Answer
The series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{0.7}} } }\) is a divergent p-series.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ n^{-\pi} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ n^{-\pi} } }\) converges or diverges using the p-series convergence theorem.
Solution
3153 video solution
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\(\displaystyle{ \sum_{n=1}^{\infty}{ (n^{-2.4} + 8n^{-1.6}) } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ (n^{-2.4} + 8n^{-1.6}) } }\) converges or diverges using the p-series convergence theorem.
Solution
3154 video solution
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\(\displaystyle{ 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} + . . . }\)
Problem Statement
Determine whether the series \(\displaystyle{ 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} + . . . }\) converges or diverges using the p-series convergence theorem.
Solution
This problem is solved by two different instructors.
3155 video solution
3155 video solution
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\(\displaystyle{ 1 + \frac{1}{\sqrt[3]{4}} + \frac{1}{\sqrt[3]{9}} + \frac{1}{\sqrt[3]{16}} + . . . }\)
Problem Statement
Determine whether the series \(\displaystyle{ 1 + \frac{1}{\sqrt[3]{4}} + \frac{1}{\sqrt[3]{9}} + \frac{1}{\sqrt[3]{16}} + . . . }\) converges or diverges using the p-series convergence theorem.
Solution
3156 video solution
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\(\displaystyle{ 1+ \frac{1}{2\sqrt[3]{2}} + \frac{1}{3\sqrt[3]{3}} + \frac{1}{4\sqrt[3]{4}} + . . . }\)
Problem Statement |
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Determine whether the series \(\displaystyle{ 1+ \frac{1}{2\sqrt[3]{2}} + \frac{1}{3\sqrt[3]{3}} + \frac{1}{4\sqrt[3]{4}} + . . . }\) converges or diverges using the p-series convergence theorem.
Final Answer |
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The series \(\displaystyle{ 1+ \frac{1}{2\sqrt[3]{2}} + \frac{1}{3\sqrt[3]{3}} + \frac{1}{4\sqrt[3]{4}} + . . . }\) is a convergent p-series.
Problem Statement
Determine whether the series \(\displaystyle{ 1+ \frac{1}{2\sqrt[3]{2}} + \frac{1}{3\sqrt[3]{3}} + \frac{1}{4\sqrt[3]{4}} + . . . }\) converges or diverges using the p-series convergence theorem.
Solution
This problem is solve by two different instructors.
PatrickJMT - 255 video solution
video by PatrickJMT |
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255 video solution
Final Answer
The series \(\displaystyle{ 1+ \frac{1}{2\sqrt[3]{2}} + \frac{1}{3\sqrt[3]{3}} + \frac{1}{4\sqrt[3]{4}} + . . . }\) is a convergent p-series.
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt[3]{n^4}} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt[3]{n^4}} } }\) converges or diverges using the p-series convergence theorem.
Solution
3158 video solution
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \sqrt[5]{n^2} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \sqrt[5]{n^2} } }\) converges or diverges using the p-series convergence theorem.
Solution
3159 video solution
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3}{\sqrt[3]{n^2}} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3}{\sqrt[3]{n^2}} } }\) converges or diverges using the p-series convergence theorem.
Solution
3160 video solution
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n^3 \sqrt{n}} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2}{n^3 \sqrt{n}} } }\) converges or diverges using the p-series convergence theorem.
Solution
3161 video solution
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\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{4\sqrt{n}}{n^4} } }\)
Problem Statement
Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{4\sqrt{n}}{n^4} } }\) converges or diverges using the p-series convergence theorem.
Solution
3162 video solution
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