Integral test

[mooshupork34]

I'm confused about how to find out whether the following series converges or diverges by the integral test, so any help would be appreciated!

sigma starting at j=2 and going to infinity of:

1/(j^4 - j)

 

[soroban]

Hello, mooshupork34!

Find whether the series converges or diverges by the integral test:

. . \(\displaystyle \L\sum_{j=2}^{\infty}\,\frac{1}{j^4\,-\,j}\)

This is a messy problem . . .

We must evaluate: \(\displaystyle \L\:\int^{\;\;\;\;\infty}_2\,\frac{dx}{x^4\,-\,x}\)

which requires Partial Fractions:

. . \(\displaystyle \L\frac{1}{x(x\,-\,1)(x^2\,+\,x\,+\,1)} \:=\:\frac{A}{x}\.+\.\frac{B}{x\,-\,1}\,+\,\frac{Cx\,+\,D}{x^2\,+\,x\,+\,1}\)


. . . . . Good luck!

 

[skeeter]

I'm confused about how to find out whether the following series converges or diverges by the integral test, so any help would be appreciated!

sigma starting at j=2 and going to infinity of:

1/(j^4 - j)

as shown by Soroban, the integral test for this series sux.

part of your study about series is learning which test(s) are best suited for determining convergence/divergence.

consider using the limit comparison test with the known convergent series 1/j⁴.