Answer to Question #136808 in Discrete Mathematics for Promise Omiponle

Question

Compute the values of the sums below.
(a)5∑(k=1) (k+ 1)
(b)4∑(j=0) (-2) ^j
(c) 10∑(i=1) 3
(d) 8∑(j=0) ( 2^(j+1) -2^j )

Answer 1

(a) $5\sum_{k=1}^{+\infty}(k+1)=5\, lim_{n\rightarrow+\infty}\sum_{k=1}^{n}(k+1)=5\, lim_{n\rightarrow+\infty}\frac{(n+2)(n+1)}{2}=+\infty$

(b) $4\sum_{j=0}^{+\infty}(-2)^j=4\, lim_{n\rightarrow+\infty}\sum_{j=0}^{n}(-2)^j\, =lim_{n\rightarrow+\infty}\frac{(-2)^{n+1}-1}{-3}=\infty$

(c) $10\sum_{i=1}^{+\infty}3=+\infty$

(d) $8\sum_{j=0}^{+\infty}(2^{j+1}-2^{j})=8\,\,lim_{n\rightarrow+\infty}\sum_{j=0}^{n}(2^{j+1}-2^{j})=$

$=8\,\,lim_{n\rightarrow+\infty}(2^{n+1}-1)=+\infty$