# Power Series Question

• December 1st 2009, 09:45 PM
bambamm
Power Series Question
A function f is defined by
f(x) = 2 + 3x + 2x2 + 3x3 + 2x4 + ...
that is, its coefficients are c2n = 2 and c2n+1 = 3 for all n=>0.
a.) Find the interval of convergence of the series
b.) Find an explicit formula for f(x).

I don't have the slightest clue how to do this problem as the representation of the series is confusing. Could somebody be to kind as to assist me on this. Thank you very much!
• December 1st 2009, 10:01 PM
qmech
Rewrite the function
$
f(x) = 2 + 3x + 2x^2 + 3x^3 + 2x^4...
$

$
f(x) = 2 + 2x + 2x^2 + 2x^3 + 2x^4...
+ x + x^3 + x^5 + ...
$

$
f(x) = 2(1 + x + x^2 + x^3 + x^4...)
+ (x + x^3 + x^5 + ...)
$

$
f(x) = 2g(x)+ h(x)
$

where
$
g(x) = 1 + x + x^2 + x^3 + x^4...
$

$h(x)= x + x^3 + x^5 + ...$

Look familiar?