## 17Calculus Infinite Series - Practice Problem 2439

Infinite Series Practice Problem 2439

$$\displaystyle{ \sum_{k=1}^{\infty}{ \left[ \frac{1}{k+1} - \frac{1}{k+2} \right] } }$$

Determine the convergence or divergence of the series.
If the series converges,
- determine the value the series converges to, if possible; and
- determine if the series converges absolutely or conditionally.

Choosing Which Tests To Try

Because of the presence of two terms with a negative sign between them, our first thought is that we might have a telescoping series. So we will try this before going on.

Applying The Tests

### Telescoping Series

This series looks like a telescoping series. To see if that is what we have, we will build a table containing the first few values and see if we can get some cancellation.

k

ak

1

$$\displaystyle{ \frac{1}{2} - \frac{1}{3} }$$

2

$$\displaystyle{ \frac{1}{3} - \frac{1}{4} }$$

3

$$\displaystyle{ \frac{1}{4} - \frac{1}{5} }$$

.
.
.

n-1

$$\displaystyle{ \frac{1}{n} - \frac{1}{n+1} }$$

n

$$\displaystyle{ \frac{1}{n+1} - \frac{1}{n+2} }$$

We can see from this table that the first term in each row is canceled by the second term in the previous row. This means that if we add all the terms together from $$k=1$$ to $$k=n$$ we will have $$1/2$$ from the first row and $$-1/(n+2)$$ in the last row, giving us the partial sum $$\displaystyle{ S_n = \frac{1}{2} - \frac{1}{n+2} }$$.

Taking the limit of $$S_n$$ will give us the value to which the series converges.
$$\displaystyle{ \lim_{n\to\infty}{S_n} = }$$ $$\displaystyle{ \lim_{n\to\infty}{\left[ \frac{1}{2} - \frac{1}{n+2}\right]} = }$$ $$\displaystyle{ \frac{1}{2} - 0 = \frac{1}{2} }$$

Therefore the telescoping series converges to $$1/2$$.

test/series works? yes best test

The telescoping series $$\displaystyle{ \sum_{k=1}^{\infty}{ \left[ \frac{1}{k+1} - \frac{1}{k+2} \right] } }$$ converges to $$1/2$$.

Notes

Due to the nature of the $$a_n$$ terms, it was not necessary to try any other test. We may have been able to use other tests to determine convergence but they would not have given us the value to which the series converges.

Really UNDERSTAND Calculus

 all infinite series topics L'Hopitals Rule

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.