For all z-values for which it converges, the function f is...

For all z-values for which it converges, the function $f$ is defined by the series
$f(z)=5+5\left(\frac{z-3}{7}\right)+5 \cdot\left(\frac{z-3}{7}\right)^{2}+5 \cdot\left(\frac{z-3}{7}\right)^{3}+\cdots$
(a) Find $f(4)$.
(b) Find the interval of convergence of $f(z)$.

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Key Concepts

Geometric Series

A geometric series is a sum of the form ? a·r^n, where ‘a’ is the initial term and ‘r’ is the common ratio between successive terms. Its sum can be expressed as a/(1 - r) provided that the absolute value of the ratio is less than one (|r| < 1), ensuring convergence. Recognizing an expression as a geometric series simplifies the process of summing the series when the ratio condition is met.

Interval of Convergence

The interval of convergence is the set of all values for which a power series converges to a finite result. For a geometric series of the form ? a·r^n, convergence is guaranteed when |r| < 1. In the context of power series, this interval is determined by establishing the radius of convergence, which is based solely on the condition imposed by the series' common ratio.

Substitution in Series Evaluation

Substitution in the context of series evaluation refers to the process of replacing the variable in a power series with a specific value to compute the function's value at that point. It is essential to confirm that the value lies within the interval of convergence so that the evaluated series represents a valid, convergent sum.

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