Prove that for all n $(-1)^1[nC1(1+rln10)/(1+ln10^n)^r1] +(-1)^2[nC2(1+rln10)/(1+ln10^n)^r]+....=0 $

Question

Prove that for nbelongs to natural number

$$(-1)^1{n\choose1}\dfrac{(1+r\ln10)}{(1+\ln(10^n))^r} +(-1)^2{n\choose2}\dfrac{(1+r\ln10)}{(1+\ln(10^n))^r}+....=0 $$

I have proved this by induction which clearly is not the method

Answer 1

Consider the binomial theorem $$(1+x)^n = 1 + \binom{n}{1} x + ... \binom{n}{n} x^n$$

But $x = -1$, we obtain:

$$(-1)^0{n\choose0}+(-1)^1{n\choose1} +(-1)^2{n\choose2}+... (-1)^n{n\choose n} =0$$

Therefore your series evaluates to $-\frac{1+r\ln{10}}{(1+\ln{10})^r}$ for all $n$ (Naturals).