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Representing functions by power series
Chapter 8, Problem 52E(choose chapter or problem)
Representing functions by power series Identify the functions represented by the following power series.
\(\sum_{k=0}^{\infty} 2^{k} x^{2 k+1}\)
Questions & Answers
QUESTION:
Representing functions by power series Identify the functions represented by the following power series.
\(\sum_{k=0}^{\infty} 2^{k} x^{2 k+1}\)
ANSWER:Solution 52E
Step 1:
In this problem we need to identify the function represented by the power series .
Consider , =
+
+
+
+.........
= +
+
+
+.......... Is an infinite geometric series , here “ first term (a) = x , common ratio (r)=
=
|
We know that a+ar+ Then (
|
Therefore , +
+
+
+.......... is an infinite geometric series, then
=
= f(x) .
Therefore , the function f(x) = represented by the power series
.