Partial sums of divergent series

Question

Let $\sum a_n$ be a divergent serie of positive terms. Prove that for each positive integer $m$ there is $n>m$ such that

$$a_{m+1}+\cdots a_n> a_1+\cdots + a_{m}.$$

I tried to use that the partial sum sequence is not Cauchy but unsuccessful.

score 2 · Accepted Answer · answered Aug 08 '19 at 14:10

2

Let $N = a_1 + \cdots + a_m$. Is there ever a partial sum that exceeds $2N$?

answered Aug 08 '19 at 14:10

Eric Towers

score 2 · Answer 2 · answered Aug 08 '19 at 14:30

2

Assume the contrary. Then the sequence $s_k=\sum_{j=1}^k a_{n+j}$ is bounded amd increasing.

answered Aug 08 '19 at 14:30

ajotatxe

Could you give more detaisl? – MotaK Aug 08 '19 at 16:18
Bounded and increasing sequences converge. Say that $s_n$ converges to $s$. Then the series converges to $a_1+a_2+\cdots+a_n+s$. – ajotatxe Aug 08 '19 at 17:08

2 Answers2