The value of the sum displaystyle∑ k =1∞ displaystyle∑ n =1∞ ( k /2 n + k ) is equal to

Question

The value of the sum displaystyle∑ k =1∞ displaystyle∑ n =1∞ ( k /2 n + k ) is equal to (A) 5 (B) 4 (C) 3 (D) 2

Tardigrade Admin · Accepted Answer

displaystyle∑ k =1∞ ( k /2 k ) displaystyle∑ n =1∞ (1/2 n )= displaystyle∑ k =1∞ ( k /2 k )[(1/2)+(1/22)+(1/23)+ ldots ldots . . . ∞]= displaystyle∑ k =1∞ ( k /2 k )=2

Q. The value of the sum $k = 1 \sum \infty n = 1 \sum \infty \frac{k}{2 ^{n + k}}$ is equal to

5

4

3

2

$k = 1 \sum \infty \frac{k}{2 ^{k}} n = 1 \sum \infty \frac{1}{2 ^{n}} = k = 1 \sum \infty \frac{k}{2 ^{k}} [\frac{1}{2} + \frac{1}{2 ^{2}} + \frac{1}{2 ^{3}} + \dots\dots ...\infty] = k = 1 \sum \infty \frac{k}{2 ^{k}} = 2$

Q. The value of the sum k=1∑∞​n=1∑∞​2n+kk​ is equal to

5

4

3

2

k=1∑∞​2kk​n=1∑∞​2n1​=k=1∑∞​2kk​[21​+221​+231​+……...∞]=k=1∑∞​2kk​=2

Q. The value of the sum $k = 1 \sum \infty n = 1 \sum \infty \frac{k}{2 ^{n + k}}$ is equal to

$k = 1 \sum \infty \frac{k}{2 ^{k}} n = 1 \sum \infty \frac{1}{2 ^{n}} = k = 1 \sum \infty \frac{k}{2 ^{k}} [\frac{1}{2} + \frac{1}{2 ^{2}} + \frac{1}{2 ^{3}} + \dots\dots ...\infty] = k = 1 \sum \infty \frac{k}{2 ^{k}} = 2$