Question

Question on series to follow:

Answer 1

If
\[a=b-\frac{b ^{2}}{2}+\frac{b ^{3}}{3}-\frac{b ^{4}}{4}+........\]
express b in ascending powers of a.

Answer 2

a = ln(b+1)
b = -1 + e^a
b = -1 + 1 + a + a^2/2! + a^3/3! + ..........
b = a + a^2/2! + a^3/3! + ..........


Derivation :
For infinite G.P.
1 - b + b^2 - b^3 + .......... = 1/(1+b) For -1<=b<=1
integrate both sides
Get ln(1+b) = b - b^2/2 + .....