Finding the Function and Interval of Convergence for the Series: Sum of ((x - 5)^(2k))/(36^k) from k = 0 to Infinity

In the field of mathematical analysis, the study of infinite series is a fundamental concept. One such series is the sum of ((x - 5)^(2k))/(36^k) from k = 0 to infinity. In this article, we will learn how to find the function represented by this series and determine the interval of convergence.

Step 1: Identifying the Series

The given series is:

Sum of ((x - 5)^(2k))/(36^k) from k = 0 to infinity

This is a power series, where the general term is ((x - 5)^(2k))/(36^k).

Step 2: Finding the Function Represented by the Series

To find the function represented by the series, we need to determine the pattern of the coefficients and the exponents of the variable 'x'.

The coefficient of the general term is 1/(36^k), and the exponent of (x - 5) is 2k.

We can rewrite the series as:

Sum of (x - 5)^(2k) / (36^k) from k = 0 to infinity

This series can be expressed as a power series in (x - 5) with the following form:

Sum of a_k(x - 5)^k from k = 0 to infinity

where the coefficients a_k are given by:

a_k = 1 / (36^k)

Therefore, the function represented by the given series is:

f(x) = Sum of (x - 5)^(2k) / (36^k) from k = 0 to infinity

Step 3: Determining the Interval of Convergence

To find the interval of convergence for the series, we can use the ratio test.

The ratio test states that if the limit of the ratio of consecutive terms in the series is less than 1, then the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.

Let's calculate the ratio of consecutive terms:

(x - 5)^(2(k+1)) / (36^(k+1)) / ((x - 5)^(2k) / (36^k))
= (x - 5)^2 / 36

The limit of this ratio as k approaches infinity is:

lim (x - 5)^2 / 36 = (x - 5)^2 / 36

The series converges if |(x - 5)^2 / 36| < 1, which is equivalent to:

-6 < x - 5 < 6

or

-1 < x < 11

Therefore, the interval of convergence for the given series is:

-1 ≤ x ≤ 11

In conclusion, the function represented by the series is:

f(x) = Sum of (x - 5)^(2k) / (36^k) from k = 0 to infinity

and the interval of convergence is:

-1 ≤ x ≤ 11