What is an example of a telescoping series and how do you find its sum?

Answer 1

A telescopic serie is a serie which can be written

#sum_{k=0}^n (a_{k+1}-a_k)#
This sum is equal to #a_{n+1}-a_0# because
#sum_{k=0}^n (a_{k+1}-a_k) = (a_1-a_0) + (a_2-a_1) + \ldots + (a_{n+1}-a_n)#.
An easy example is #sum_{k=1}^infty 1/(n(n+1))#.
Remark that #1/(n(n+1)) = 1/n - 1/(n+1)#, so,
#sum_{k=1}^N 1/(n(n+1)) = (1-1/2) + (1/2 - 1/3) + \ldots + (1/N - 1/(N+1))#
#sum_{k=1}^N 1/(n(n+1)) = 1 - 1/(N+1)#.
When #N -> +infty#, you get #sum_{k=1}^infty 1/(n(n+1))=1#.
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Answer 2

An example of a telescoping series is the series (\sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right)). To find the sum of a telescoping series, you can use the method of partial fraction decomposition to simplify the series into a form where most terms cancel each other out, leaving only a finite number of terms. Then, you can evaluate the remaining terms to find the sum. For this example, the partial fraction decomposition simplifies the series to (\sum_{n=1}^{\infty} \frac{1}{n} - \frac{1}{n+1} = 1), as most terms cancel out, leaving only the first term (\frac{1}{1}). Therefore, the sum of the telescoping series is 1.

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Answer 3

An example of a telescoping series is ( \sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right) ). To find its sum, you can simplify the expression by expanding it and then grouping terms together. This will result in many terms canceling each other out, leaving only a finite number of terms. Finally, you sum these remaining terms to find the total sum of the series.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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