Question

Provide an example of a series such that the partial sums S, are bounded but the series . k diverges.

Answer 1

To determine the example of a series such that the partial sums S, are bounded but the series . k diverges.

Provide an example of a series

`\Sigma^(\infty)_(k=1)`

such that the partial sum Sn are bounded but the series

`\Sigma^(\infty)_(k=1)a_{k\text{ }}\text{diverge}`

Provide an example of a geometric series

`\Sigma^(\infty)_(k=0)r^k`

such that

`\Sigma^(\infty)_(k=0)r^k\\e(1)/(1-r)`

Provide an example of a series

`\Sigma a_k`

such that

`\Sigma a_k`

diverges but

`\lim _(k\to\infty)a_k=0`