Power Series?

Question

Find the sum of the series
a)1-ln2+((ln2)^2)/(2!)-((ln2)^3)/(3!)+....
b)sum(3^n/(5^n(n!))) n=0 to infinity

Answer 1

a: 1/2. Note that e^x=1+x+x²/2!+x³/3!+x^4/4!... and substituting -ln 2 for x gives the original series, thus the sum of the series is e^(-ln 2). Rearranging terms gives you e^(ln (2^(-1))), which is 2^(-1), which is 1/2.

b: This is also the taylor series of e^x, this time with x=3/5, so the sum is e^(3/5), which is approximately 1.82212.