# Thread: To find the exact value of a series

1. ## To find the exact value of a series

A sequence is defined by u(1)=0 and (1+n) * u(n+1) = n + u(n) for positive integer n. Prove that u(n) = 1 - 1/(n!).
Hence, find the exact value of the sum to infinity of (1 - u(r))/(2^r).

I can prove u(n) = 1 - 1/(n!) by mathematical induction.
May I know how to answer the "hence" part?

2. ## Re: To find the exact value of a series

Originally Posted by acc100jt
A sequence is defined by u(1)=0 and (1+n) * u(n+1) = n + u(n) for positive integer n. Prove that u(n) = 1 - 1/(n!).
Hence, find the exact value of the sum to infinity of (1 - u(r))/(2^r).

I can prove u(n) = 1 - 1/(n!) by mathematical induction.
May I know how to answer the "hence" part?
If $u_{n}= 1-\frac{1}{n!}$ then $1-u_{n}=\frac{1}{n!}$ , so that the 'infinite sum' is...

$\sum_{n=1}^{\infty} \frac{(\frac{1}{2})^{n}}{n!}= \sum_{n=0}^{\infty} \frac{(\frac{1}{2})^{n}}{n!} -1= \sqrt{e}-1$ (1)

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