Question

use the integral test to determine if the series shown below converges or diverges. series from 1 to infinity 2n^2/e^(n/2)

Answer 1

what are the assumptions needed for the integral test?

Answer 2

suppose that the f is a continuous, positive, decreasing function on (1,infinity). if f(n)=a for n greater or equal to 1, then \[\sum_{1}^{\infty} a_n and \int\limits_{1}^{\infty} f(x)dx\] either both converge or diverge

Answer 3

ok so first we need to show that 2n^2/e^(n/2) is positive and decreasing

of course its positive because n is natural
so now we take the derivative to check if f(x) is decreasing

Answer 4

note that if it decreases on R then of course it decreases on N

Answer 5

can you take the derivative of f(x) = 2x^2/e^(x/2) ?

Answer 6

@liz.scarce

Answer 7

you use integration by parts \[-4n^2/e^{n/2}-16/e ^{n/2}+\int\limits_{4}^{\infty} 16dn/e ^{n/2}\]

Answer 8

but first you need to show that a_n is decreasing

Answer 9

then yes, take the integral and if you get a finite number then your series converges