Convergence of Series Using Integral Test

[clairez93]

Homework Statement

Use the integral test to determine the convergence or divergence of the series.

\Sigma^{\infty}_{n=1}\frac{n^{k-1}}{n^{k}+c} k is a positive integer

Homework Equations

The Attempt at a Solution

Consider:

\int^{/infty}_{1}\frac{x^{k-1}}{x^{k}+c} dx

Not sure how to integrate this expression to determine if it converges or diverges.

 

[zcd]

This is more of a guess on my part, but since the degree of the polynomial on the denominator is only one greater than that of the numerator, the answer will be the form of natural log (i.e.\int \frac{x^0}{x^1} dx = \ln_|x| + C). That means the integral is divergent -> series is divergent. You could also check for divergence with the limit comparison test, with a divergent series such as 1/n.

 

[g_edgar]

Consider:

\int^{\infty}_{1}\;\frac{x^{k-1}}{x^{k}+c} dx

Not sure how to integrate this expression to determine if it converges or diverges.

What u-substitution will make this integral easy?

 

[Bohrok]

Try letting u be the denominator and make du look like the numerator in the integrand.