determine-tbe-sewuence-u-n-wich-verify-u-n-u-n-1-1-n-n-n-gt-0-

Question Number 124065 by Bird last updated on 30/Nov/20

d e t e r m i n e t b e s e w u e n c e u_{n}

w i c h v e r i f y u_{n} + u_{n + 1} = \frac{{(- 1)}^{n}}{\sqrt{n}}

n > 0

Answered by mindispower last updated on 30/Nov/20

{(- 1)}^{n} u_{n} + {(- 1)}^{n} u_{n + 1} = \frac{1}{\sqrt{n}}

\underset{k = 1}{\sum^{n}} {(- 1)}^{k} u_{k} + {(- 1)}^{k} u_{k + 1} = \underset{k = 1}{\sum^{n}} \frac{1}{\sqrt{k}}

{(- 1)}^{n} u_{n + 1} - u_{1} = \underset{k = 1}{\sum^{n}} \frac{1}{\sqrt{k}}

u_{n} = {(- 1)}^{n - 1} u_{1} + {(- 1)}^{n - 1} \sum_{k ⩽ n - 1} \frac{1}{\sqrt{k}}, \forall n ⩾ 2