Whether an infinite series can be tested by integral test

Question

I am asked whether the following infinite series can be proved to be convergent by integral test.

$$\sum_{n=1}^\infty n e^{6 n}$$

so I integrate it

$$\int_1^{\infty}\ n e^{6n}\, dn$$

and find it diverges so I concluded that the above series also diverges by the integral test. However, the answer is that the integral test cannot be used to test this infinite series. What is wrong in my deduction?

See for the main theorem for integral test. – Mhenni Benghorbal May 15 '13 at 11:36 — Mhenni Benghorbal, May 15 '13 at 11:36

score 2 · Accepted Answer · answered May 15 '13 at 11:17

2

Hint:

Is the sequence $\,\{ne^{6n}\}_{n\in\Bbb N}\,$ monotone decreasing?

answered May 15 '13 at 11:17

DonAntonio

214,715

score 1 · Answer 2 · answered May 15 '13 at 11:17

1

Look carefully at the statement of the integral test. I suppose that $n \mapsto n e^{6n}$ should be decreasing, which is clearly false.

By the way, since $\lim_{n \to \infty} n e^{6n} = +\infty$, how can the series converge?

answered May 15 '13 at 11:17

Siminore

35,786

Whether an infinite series can be tested by integral test

2 Answers2