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Under what condition will an infinite geometric series \sum_{n=1}^{\infty}a.r^{n-1} converge? We...

Question:

Under what condition will an infinite geometric {eq}\sum_{n=1}^{\infty}a.r^{n-1} {/eq} series converge?

We wish to show whether {eq}\sum_{n=1}^{\infty}a_{n} {/eq} converges by using the Integral Test. So we choose an f(x) with {eq}f(n)=a_{0} {/eq} over {eq}(1,\infty) {/eq}. What three things must be true about f(x)?

Integral test:

The integral test is a test to know the convergence or divergence of a series by defining some function {eq}f {/eq} such that {eq}f {/eq} is positive, decreasing and continuous for {eq}x > c {/eq}, then if the integral converges means the series converges as well.

Answer and Explanation:

From the definition of Geometric Series Convergence, the series in the form of {eq}\sum_{n=1}^{\infty}a.r^{n-1} {/eq} will be converge when {eq}-1 < r < 1 {/eq} which can also be written as {eq}|r| < 1 {/eq}.

From the definition of Integral test, for {eq}n \geq 1 {/eq} and {eq}f(n) {/eq}, then {eq}f {/eq} should be positive, continuous and decreasing.


Learn more about this topic:

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Infinite Series & Partial Sums: Explanation, Examples & Types

from GRE Math: Study Guide & Test Prep

Chapter 12 / Lesson 4
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