Question

(15-16) Consider the Infinite Geometric Series:

a) Find the partial sums S_n for n = 1, 2, 3, and 4.

b) Does the series have a sum?​

Answer 1

15 Answer: S₁ = 1 S₂ = 4 S₃ = 13 S₄ = 40 Sum = NO

Explanation:

1 + 3 + 9 + 27 + ...
`\implies\sum^(\infty)_(n=1)3^(n-1)\implies\sum^(\infty)_(n=1)(3^n)/(3)\\\\\bullet S_1=1\\\bullet S_2=1+3=4\\\bullet S_3=1+3+9=13\\\bullet S=1+3+9+27=40\\\\\\ \lim_(n \to \infty) (3^n)/(3) \implies(3^(\infty))/(3)\implies\infty\\\\\text{The series diverges so there is no sum.}`

16 Answer:
`\bold{S_1=(1)/(2)\qquad S_2=(2)/(3)\qquad S_3=(13)/(18)\qquad S_4=(39)/(54)\qquad Sum=YES}`

Explanation:

`(1)/(2)+(1)/(6)+(1)/(18)+(1)/(54)+(1)/(162)+...\implies \sum^(\infty)_(n=1)(1)/(2)\bigg((1)/(3)\bigg)^(n-1)\\\\\\\bullet S_1=(1)/(2)\\\\\bullet S_2=(1)/(2)+(1)/(6)=(2)/(3)\\\\\bullet S_3=(1)/(2)+(1)/(6)+(1)/(18)=(13)/(18)\\\\\bullet S_4=(1)/(2)+(1)/(6)+(1)/(18)+(1)/(54)=(39)/(54)`

`\lim_(n \to \infty) (1)/(2)\bigg((1)/(3)\bigg)^(n-1)\implies (1)/(2)\lim_(n \to \infty) (1)/(3^(\infty-1))\implies (1)/(\infty)=0\\\\\\\text{The series converges so it does have a sum.}`