Mathematics Asked by Ramana on December 5, 2020

Let

$$S = -int_{1}^{2}frac{1}{te^t},dt + int_{2}^{3}frac{1}{te^t},dt-int_{3}^{4}frac{1}{te^t},dt + cdots +text{ad inf}$$

Does the series $S$ converge? Clearly,

$$S=sum_{ngeq 1}(-1)^{n}int_{n}^{n+1}frac{1}{t e^t},dt$$

So I thought of using alternating series test as there is a $(-1)^{n}$ term, but I am not sure how to estimate the integral term.

Notice that $1/te^t$ is a monotone decreasing function as $t$ grows. Thus (readily evident from the interpretation of an integral as signed area),

$$int_{n}^{n+1} frac{1}{te^t} dt le frac{1}{ne^n}$$

I believe this should be enough to get you to the desired conclusion.

Correct answer by Eevee Trainer on December 5, 2020

Set $$b_n=int_{n}^{n+1}frac{1}{t e^t},dt,$$ then $$frac{1}{(n+1)e^{n+1}}leq b_nleqfrac{1}{ne^{n}},$$ and using squeeze theorem, $$lim_{nto +infty}{b_n}=0.$$

Obviously, ${b_n}$ is decreasing, therefore $S=sum_n{(-1)^nb_n}$ is convergent.

Answered by Noah Tang on December 5, 2020

The function begin{equation} f(t)=frac{1}{te^t} end{equation} is continuous, strictly decreasing on $[1,+infty)$ and its limit as $tto+infty$ is $0$. This is enough to conclude that the terms begin{equation} a_n=int_n^{n+1}frac{1}{te^t},text{d}t end{equation} are decreasing and converge to $0$ as $nto+infty$. Indeed, by the mean value theorem, there are $t_nin[n,n+1]$ and $t_{n+1}in[n+1,n+2]$ such that $a_n=f(t_n)$ and $a_{n+1}=f(t_{n+1})$, thus $a_n=f(t_n)>f(t_{n+1})=a_{n+1}$ and $a_n=f(t_n)to 0$ as $nto+infty$. All the requirements of the alternating series test are therefore fulfilled and allow to conclude that the series converges.

Answered by Davide Ravasini on December 5, 2020

Get help from others!

Recent Answers

- Jon Church on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- haakon.io on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP