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Q37E

Expert-verifiedFound in: Page 443

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

**Find the values of x for which the series converges. Find the sum of the series for those values of x.**

\(\sum\nolimits_{n = 0}^\infty {\frac{{{{(x - 2)}^n}}}{{{3^n}}}} \)

The given series converges for \( - 1 < x < 5\). Sum of series for these values of \(x = \frac{3}{{5 - x}}\).

**Finding the interval where the series converges by using the concept that an infinite geometric whose **\(|r| < 1\)** is convergent.**

This series converges if we assume it to be of the form of an infinite geometric series, \(\sum\limits_{k = 0}^\infty {a{r^k} = \frac{a}{{1 - r}},|r| < 1} \)

Comparing the given series with the infinite geometric series, we observe that \(a = 1\) and \(r = \frac{{x - 2}}{3}\).

Putting |r|<1, \(|\frac{{x - 2}}{3}| < 1\)

ie \( - 1 < \frac{{x - 2}}{3} < 1\) \( \Rightarrow - 3 < x - 2 < 3 \Rightarrow - 1 < x < 5.\)

Therefore, the series converges when \(x\) belongs to \(( - 1,5).\)

Sum of an infinite geometric progression is given by \(\frac{a}{{1 - r}}\).

Finding sum of series when \(x\) is in the range \(( - 1,5).\)

So, we obtain:\(\frac{1}{{1 - (\frac{{x - 2}}{3})}} = \frac{3}{{5 - x}}\).

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