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GRE Subject Test: Math : Comparing Rates of Convergence

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Example Questions

Example Question #121 : Convergence And Divergence

For which values of p is

$\sum_{n=1}^\infty \frac{1}{n^p}$ $\displaystyle \sum_{n=1}^\infty \frac{1}{n^p}$

convergent?

Possible Answers:

p=2 $\displaystyle p=2$

All positive values of $\displaystyle p$

it doesn't converge for any values of $\displaystyle p$

only p > 1 $\displaystyle p > 1$

Correct answer:

only p > 1 $\displaystyle p > 1$

Explanation:

We can solve this problem quite simply with the integral test. We know that if

$\int_1^{\infty} \frac{1}{x^p} dx$ $\displaystyle \int_1^{\infty} \frac{1}{x^p} dx$

converges, then our series converges.

We can rewrite the integral as

$\int x^{-p} dx$ $\displaystyle \int x^{-p} dx$

and then use our formula for the antiderivative of power functions to get that the integral equals

$\frac{x^{1-p}}{1-p} \Big|_{x=1}^\infty$ $\displaystyle \frac{x^{1-p}}{1-p} \Big|_{x=1}^\infty$.

We know that this only goes to zero if 1-p < 0 $\displaystyle 1-p < 0$. Subtracting p from both sides, we get

1 < p $\displaystyle 1 < p$.

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GRE Subject Test: Math : Comparing Rates of Convergence

Study concepts, example questions & explanations for GRE Subject Test: Math

All GRE Subject Test: Math Resources

Example Questions

Example Question #121 : Convergence And Divergence

All GRE Subject Test: Math Resources

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