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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?

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Hint:

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

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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?

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