Question

A function f is defined by f(x) = 1 + 6x + x2 + 6x3 + x4 + ⋯ that is, its coefficients are c2n = 1 and c2n + 1 = 6 for all n ≥ 0. Find the interval of convergence of the series. Find an explicit formula for f(x).

Answer 1

From the odd-degree terms, take out one copy and rewrite the series as

$1+6x+x^2+6x^3+\cdots=(1+x+x^2+x^3+\cdots)+5x+5x^3+\cdots$

$1+6x+x^2+6x^3+\cdots=(1+x+x^2+x^3+\cdots)+5x(1+x^2+\cdots)$

Then if |x| < 1, we can condense this to

$\displaystyle\sum_{n=0}^\infty x^n+5x\sum_{n=0}^\infty x^{2n}=\frac1{1-x}+\frac{5x}{1-x^2}=\frac{1+6x}{1-x^2}$

Since the series we invoked here converge on -1 < x < 1, so does this one.