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Find the values of x for which the series \sum_{n=0}^\infty \frac{(2x-3)^n}{2^n} converges.

Question:

Find the values of {eq}x {/eq} for which the series {eq}\sum_{n=0}^\infty \frac{(2x-3)^n}{2^n} {/eq} converges.

Sum of a Series:

The series with the exponent variable, as in the case of the given problem, we will be using the root test. Now, to apply the root test, we have {eq}\mathrm{If\:}\lim _{n\to \infty }|a_n|^{\frac{1}{n}}=L\mathrm{,\:and:}\\ \mathrm{If\:}L>1\mathrm{,\:then\:}\sum a_n\mathrm{\:diverges}\\ \mathrm{If\:}L<1\mathrm{,\:then\:}\sum a_n\mathrm{\:converges} {/eq}

Answer and Explanation:


In the given series, which is:

{eq}\sum_{n=0}^\infty \frac{(2x-3)^n}{2^n}\\ {/eq}

we will apply the root test. As per the root tests, we have:

{eq}\lim _{n\to \infty }|a_n|^{\frac{1}{n}}\\ =\lim _{n\to \infty \:}\left(\left|\left(\frac{\left(2x-3\right)^n}{2^n}\right)^{\frac{1}{n}}\right|\right)\\ \left|\frac{2x-3}{2}\right|\\ {/eq}

Now we have the condition:

{eq}\Rightarrow \frac{1}{2}<x<\frac{5}{2}\\ {/eq}

In the interval the end points are not included, as it will diverge the series.


Learn more about this topic:

Using Sigma Notation for the Sum of a Series

from Algebra II Textbook

Chapter 21 / Lesson 13
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