For a convergent alternating series [tex]\sum\limits_{n=1}^\infty(-1)^{n+1}a_n[/tex] with value [tex]S[/tex] and [tex]k[/tex]th partial sums denoted by [tex]S_k[/tex], the [tex]k[/tex]th error is bounded by the absolute value of the [tex](k+1)[/tex]th term's absolute value:
[tex]|S-S_k|\le|a_{k+1}|[/tex]
We have
[tex]a_n=\dfrac1{n^7}[/tex]
so in order to have an error within 0.00005 of the sum's actual value, we need [tex]k[/tex] terms such that
[tex]\left|S-S_k\right|\le\left|\dfrac{(-1)^{k+2}}{(k+1)^7}\right|=\dfrac1{(k+1)^7}<0.00005[/tex]
[tex]\implies (k+1)^7<\dfrac1{0.00005}=20000[/tex]
[tex]\implies k+1<20000^{1/7}\approx4.1156[/tex]
[tex]\implies k<3.1156[/tex]
which suggests that we require [tex]k=4[/tex] terms at the least to approximate the series within the given accuracy.
