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I need some help with the following question from my homework. Any and all help is appreciated. Problem at hand Thank you all.

Edit: The question reads as follows:

Let $\{x_n\}$ be a convergent sequence with limit $x$. Let $y_n=x_n-x_{n+1}$ for all positive integers $n$. Evaluate $\sum_{n=1}^\infty y_n$, if the infinite series converges.

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Could you add the question to the body of the text and not just some external image? – Jonas Teuwen Oct 18 '11 at 22:01
up vote 5 down vote accepted

Look at the partial sums. We have $$S_N = \displaystyle \sum_{n=1}^{N} y_n = \displaystyle \sum_{n=1}^{N} (x_n - x_{n+1}) = (x_1 - x_2) + (x_2 - x_3) + \cdots + (x_N - x_{N+1}) = x_1 - x_{N+1}$$ Hence, $$\displaystyle \sum_{n=1}^{\infty} y_n = \lim_{N \rightarrow \infty} S_N = \lim_{N \rightarrow \infty} (x_1 - x_{N+1}) = x_1 - x$$

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Hint: This series telescopes. We have $$\sum_{n=1}^\infty y_n = y_1 +y_2 +y_3 +y_4 +y_5 +\cdots =(x_1-x_2)+(x_2-x_3)+(x_3-x_4)+(x_4-x_5)+\cdots$$ Taking the partial sum up to $x_N$, we have that $$= x_1+ (-x_2+x_2)+(-x_3+x_3)+(-x_4+x_4)+\cdots+(-x_N+x_N)-x_{N+1}=x_1-x_{N+1}.$$

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