# Integral test and series

1. Jul 3, 2009

### clairez93

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jul 3, 2009

### 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.

Last edited: Jul 3, 2009
3. Jul 3, 2009

### g_edgar

What u-substitution will make this integral easy?

4. Jul 3, 2009

### Bohrok

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