$a_n=\sum\nolimits_{i=0}^n 1/(n+i)$ I do not know how to manipulate this

Question

$a_n=\sum\nolimits_{i=0}^n 1/(n+i)$ I have to prove that the series converges and the limit. but i dont have a clue of how to approach it. It seems familiar to $log(1+x)$ but since $n$ is in the denominator i have difficulties to manipiulate it. Thanks in advance

do i need to use integrals ? we hadnt integrals so far, i have been doing some old exam papers. — Maths, Jan 31 '20 at 00:03
Not all answers in the linked question use integrals, have you looked at them? — YiFan Tey, Jan 31 '20 at 00:04

score 0 · Answer 1 · answered Jan 31 '20 at 00:00

0

Hint:

Rewrite it as $$ \frac1n \sum_{i=1}^n \frac1{1+\frac in}$$ and observe this is a Riemann sum.

answered Jan 31 '20 at 00:00

Bernard

175,478

we hadnt riemann sums :/ – Maths Jan 31 '20 at 00:00
1

In this case, it will be hard, because the limit is $\ln 2$, which supposes you've seen the integral. – Bernard Jan 31 '20 at 00:03

$a_n=\sum\nolimits_{i=0}^n 1/(n+i)$ I do not know how to manipulate this

1 Answers1