let-U-n-p-k-1-n-1-n-k-p-1-with-n-p-from-N-1-calculate-lim-n-U-n-p-for-p-2-2-prove-that-U-n-1-is-convergent-3-let-V-n-k-1-n-sin-1-n-k-2-find-lim-V

Question Number 29503 by abdo imad last updated on 09/Feb/18

l e t U_{n, p} = \sum_{k = 1}^{n} \frac{1}{{(n + k)}^{p + 1}} w i t h n, p f r o m N^{★}

1) c a l c u l a t e {l i m}_{n \to + \infty} U_{n, p} f o r p ⩾ 2

2) p r o v e t h a t U_{n, 1} i s c o n v e r g e n t

3) l e t V_{n} = \sum_{k = 1}^{n} s i n (\frac{1}{{(n + k)}^{2}}) f i n d {l i m}_{\infty} V_{n} .