Question

1. Find an infinite series series of positive terms which sums to 3
2. Find the exact value of 1-2+4/2!-8/3!+16/4!-31/5!+64/6!-128/7!..+(-2)ⁿ/n!

Answer 1

$(1)\varSigma_{n=1}^\infin ar^{n-1}=3\\ \\−1<r<1 \ is \\ S=\frac{{a}_{1}}{1-r}$

where $a=first \ term \ \&\ r=common\ ratio$

$S=3$

$3=\frac{a}{1-r}$

therefore

$r=\frac{3-a}{3}$

Apply the condition for convergence to determine possible values of a

$For\ a\ series\ to\ converge: −1<r<1$

$−1<r<1$

$−1<\frac{3-a}{3}<1$

$−3<3-a<3$

$-6<−a<0$

$0<a<6$

$final\ answer\\ For\ the\ series\ to\ converge, 0<a<6\ and −1<r<1.$

(2) $1-2+\frac{4}{2!}-\frac{8}{3!}+\frac{16}{4!}-\frac{32}{5!}+\frac{64}{6!}-\frac{128}{7!}+--++\frac{-2^n}{n!}+--$

$=e^{-2}=0.135$

$=0.135$