For the series to converge, $R_n$ must be zero when n tends to infinity. In equation (5.101), when m is a nonnegative integer and x=1, the only effective term left is $\frac{1}{n!}$ that becomes zero when n tends to infinity. So, why is x=1 not included in the range of x for this series to converge?
In the expansion of ln(1+x) the remainder term(effective) is $\frac{x^n}{n}$ and x=1 was included in its range. Both are similar. Why is there a difference in including x=1?
Best Answer
Observe that $m(m-1)\dots(m-n+1)$ has $n$ factors (in particular depends on $n$), and $$ \frac{m(m-1)\dots(m-n+1)}{n!}\not\to0. $$