1 + 1/2 + 1/4 + 1/8 + ……………. ∞ terms = A) 3B) 2 C) 1/2D)

1.	1 + 1/2 + 1/4 + 1/8 + ……………. ∞ terms = A) 3B) 2 C) 1/2D) 1
Answer» Correct option is (B) 2 \(S_n=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.......+\) upto n terms \(=\cfrac{1(1-(\frac{1}{2})^n)}{1-\frac{1}{2}}\) \((\because S_n=\frac{a(1-r^n)}{r-1},r<1\) in G.P.) \(=\cfrac{1-\frac{1}{2^n}}{\frac{1}{2}}\) \(=2(1-\frac1{2^n})\) Sum of infinite terms \(=S_\infty=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+......(\infty-terms)\) \(=\underset{n\rightarrow\infty}{lim}\,S_n\) \(=\underset{n\rightarrow\infty}{lim}\,2(1-\frac1{2^n})\) \(=2(1-\frac1\infty)\) \(=2(1-0)=2\) Correct option is B) 2

1 + 1/2 + 1/4 + 1/8 + ……………. ∞ terms = A) 3B) 2 C) 1/2D) 1

Answer»

Correct option is (B) 2

\(S_n=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.......+\) upto n terms

\(=\cfrac{1(1-(\frac{1}{2})^n)}{1-\frac{1}{2}}\) \((\because S_n=\frac{a(1-r^n)}{r-1},r<1\) in G.P.)

\(=\cfrac{1-\frac{1}{2^n}}{\frac{1}{2}}\)

\(=2(1-\frac1{2^n})\)

Sum of infinite terms \(=S_\infty=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+......(\infty-terms)\)

\(=\underset{n\rightarrow\infty}{lim}\,S_n\)

\(=\underset{n\rightarrow\infty}{lim}\,2(1-\frac1{2^n})\)

\(=2(1-\frac1\infty)\)

\(=2(1-0)=2\)

Correct option is B) 2