let-u-n-x-1-n-x-n-n-1-dt-t-x-with-x-1-2-1-prove-that-0-u-n-x-1-n-x-1-n-1-x-n-gt-0-2-prove-that-u-n-x-converges-let-n-1-u-n-1-3-find-n-1-u-n-x-i

Question Number 62420 by mathsolverby Abdo last updated on 20/Jun/19

l e t u_{n} (x) = \frac{1}{n^{x}} - \int_{n}^{n + 1} \frac{d t}{t^{x}} w i t h x \in [1, 2]

1) p r o v e t h a t 0 ⩽ u_{n} (x) ⩽ \frac{1}{n^{x}} - \frac{1}{{(n + 1)}^{x}} (n > 0)

2) p r o v e t h a t Σ u_{n} (x) c o n v e r g e s

l e t γ = \sum_{n = 1}^{\infty} u_{n} (1)

3) f i n d \sum_{n = 1}^{\infty} u_{n} (x) i n t e r m s o f ξ (x) a n d

1 - x

4) p r o v e t h a t t h e c o n v e r g . o f Σ u_{n} (x) i s

u n i f o r m

p r o v e t h a t f o r x \in V (1)

ξ (x) = \frac{1}{x - 1} + γ + o (1)

5) f i n d t h e v a l u e o f \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1}}{n} l n (n)