Answer:
Number of terms required is 5 terms
Step-by-step explanation:
Given the series: [tex]\sum\imits^{\infty}_{n=0}\frac{(-1)^n}{5^nn!}[/tex]
Given the series is an alternating series,
[tex]b_n=\frac{1}{5^nn!}[/tex]
Evaluating the limit: [tex]\lim_{n \to \infty} b_n= \lim_{n \to \infty} (\frac{1}{5^nn!})=\frac{1}{\infty}=0[/tex]
Since [tex]\lim_{n \to \infty} b_n =0 \,\, and \,\, b_{n+1}\leq b_n[/tex] for all n = the series is convergent
The error of an alternating series [tex]\sum b_n[/tex] is bounded as
[tex]|R_n|\leq b_{n+1}[/tex]
Given [tex]b_n = \frac{1}{5^n n!}=b_{n+1}=\frac{1}{5^{n+1}(n+1)!}<0.000005\\\\=5^{n+1}(n+1)!>200000[/tex]
By trial and error: the above equation is satisfied for [tex]n=4[/tex]
Since the given series starts at n = 0, the number of terms required is 5 terms
