Question

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=1}^{\infty} \frac{(x-2)^{k}}{3^{2 k}}$$

Short answer

Question: Determine the function represented by the given series and find its interval of convergence: $$\sum_{k=1}^{\infty} \frac{(x-2)^{k}}{3^{2 k}}$$ Answer: The function represented by the series is $$S(x) = \frac{9(x-2)}{8}$$ and its interval of convergence is $$(-7, 11)$$.

Step-by-step solution

Identify the geometric series

We are given the series: $$\sum_{k=1}^{\infty} \frac{(x-2)^{k}}{3^{2 k}}$$ This is the form of a geometric series with $$a = (x-2)$$ and common ratio $$r = \frac{1}{9}$$.

Apply geometric series summation formula

The geometric series summation formula is: $$S = \frac{a}{1-r}$$ Now we can substitute the values for $$a$$ and $$r$$ from Step 1: $$S = \frac{x-2}{1-\frac{1}{9}} = \frac{(x-2)}{\frac{8}{9}}$$ This simplifies to: $$S = \frac{9(x-2)}{8}$$ Now we have found the function represented by the series: $$S(x) = \frac{9(x-2)}{8}$$

Apply the Ratio Test for convergence

The Ratio Test states that if: $$\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = L$$ If $$L < 1$$, the series converges. If $$L > 1$$, the series diverges. If $$L = 1$$, the test is inconclusive. So, let's compute the ratio $$\frac{a_{k+1}}{a_k}$$: $$\frac{a_{k+1}}{a_k} = \frac{\frac{(x-2)^{k+1}}{3^{2(k+1)}}}{\frac{(x-2)^{k}}{3^{2k}}} = \frac{(x-2)^{k+1}3^{2k}}{(x-2)^{k}3^{2(k+1)}}$$ Now, let's cancel out some terms and simplify: $$\frac{a_{k+1}}{a_k} = \frac{(x-2)}{9}$$ The limit is already independent of $$k$$, so we have: $$L = \left|\frac{x-2}{9}\right|$$ Now, we will find the interval of convergence by solving for $$x$$: When: $$L < 1 : \left|\frac{x-2}{9}\right| < 1$$ This implies: $$-9 < x-2 < 9$$ So, we get the interval $$(-7, 11)$$. The series converges for $$x$$ in the interval $$(-7, 11)$$. In conclusion, the function represented by the series is $$S(x) = \frac{9(x-2)}{8}$$ and the interval of convergence is $$(-7, 11)$$.

Key concepts

Interval of Convergence

The interval of convergence of a series is the set of all values of \(x\) for which the series converges. It essentially tells us the range over which the series provides a usable and valid function. Finding this interval is crucial when dealing with power series, as it determines where our function behaves well.

For a geometric series like the one given in our exercise, \(\sum_{k=1}^{\infty} \frac{(x-2)^k}{3^{2k}}\), we start by checking if the series converges through the use of the Ratio Test. This involves finding the common ratio \(r\) of the series. In our case, \(r = \frac{1}{9}\). Next, we focus on solving the inequality \(\left|r\right| < 1\). This will give us:

• \(\left| \frac{x-2}{9} \right| < 1\)
• \(-9 < x-2 < 9\)

By further simplifying, we determine the interval \(x\) must satisfy is \((-7, 11)\). This interval reflects the range of \(x\) values for which the series converges to a function.

Series Convergence

Series convergence refers to whether the sum of infinite terms in a series approaches a finite value. Convergence is a critical concept in calculus and analysis because it determines if a series can effectively represent a function.

For the series \(\sum_{k=1}^{\infty} \frac{(x-2)^k}{3^{2k}}\), we recognized it as a geometric series with terms that follow a pattern defined by a ratio between consecutive terms. Identifying such patterns enables us to use specific tests to determine convergence.

The series converges when the absolute value of the series' common ratio is less than one. Using the value obtained from the Ratio Test, we see that \(\left|\frac{x-2}{9}\right| < 1\). This condition must be satisfied for the series to converge. Therefore, it is vital to always confirm the common ratio of a series and use convergence tests like the Ratio Test to verify series behavior.

Ratio Test

The Ratio Test is a powerful method used for determining the convergence of a series. It is particularly useful for series with factorials, exponentials, and geometric behaviors. The Ratio Test works by examining the limit of the ratio of successive terms in a series.

In applying the Ratio Test to our series, we calculate \(a_{k+1}/a_k\), which simplifies to \(\left| \frac{x-2}{9} \right|\). This makes our limit independent of \(k\), simplifying our work as we only need to solve \(\left|\frac{x-2}{9}\right| < 1\) to determine where the series converges.

The Ratio Test states:

• If \(L < 1\), the series converges absolutely.
• If \(L > 1\), the series diverges.
• If \(L = 1\), the test is inconclusive.

In our analysis, \(\left| \frac{x-2}{9} \right| < 1\) provides us with clear guidelines as to where the function defined by the series is valid, and we obtained \((-7, 11)\) as the interval of convergence. This test simplifies the convergence analysis and assures us of the series' behavior across different value ranges of \(x\).