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Find: Use the integral test to determine whether the series is \sum _{n=1}^{\infty} ne^{-n^2} ...

Question:

Use the integral test to determine whether the series is

{eq}\displaystyle \sum _{n=1}^{\infty} ne^{-n^2} {/eq} divergent or convergent. Then, we want to approximate the sum of the same series by the partial sum {eq}s_{n} {/eq} . How large should {eq}n {/eq} be to ensure that the error of this approximation is less than {eq}10^{-4} {/eq} Leave the answer as an inequality for .

Integral test

We will use integral test to simplyfy the convergaeion of sum to convergetion of the integral. The integral is eavaluate by Newton-Libnitz formula, e.g. staraightforward computation. After that, we use the definition of series sum to find the required approximation.

Answer and Explanation:

{eq}\sum_{n=1}^{\infty} ne^{-n^2} {/eq} is convergent by integral test if and only if {eq}\int_1^{\infty} xe^{-x^2} \, dx {/eq} converges. Calculating we find

$$\int_1^{\infty} xe^{-x^2} \, dx = \frac{1}{2} \int_1^{\infty} e^{-t}dt = \frac{e^{-t}}{2} \Big|_1^{\infty} =\frac{1}{2e} < \infty $$

Hence, the sum converges.

Since {eq}a_n= 4e^{-16} \approx 4,5 \cdot 10^{-7} << 10^{-4} {/eq} we conclude that {eq}n {/eq} should be greater or equal to {eq}4 {/eq} for such an approximation.


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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