Answers

2014-08-24T20:17:15+05:30
First of all, if you will notice, it is Geometric Series, with-
(1) First Term a = 1 / 2
(2) Common ratio r = 1 / 2

Let's represent this as G(a =  \frac{1}{2}, r =  \frac{1}{2})

Sum(G)-of-first-n-terms =  \frac{a(1 -  r^{n} )}{(1 - r)} = \frac{ \frac{1}{2} (1 -  ( \frac{1}{2} )^{n} )}{(1 -  \frac{1}{2} )} = \frac{ \frac{1}{2} (1 -  ( \frac{1}{2} )^{n} )}{(\frac{1}{2} )} \\  \\ = 1 -  ( \frac{1}{2} )^{n}

Now if \lim_{n \to \infty} Sum(G)(n) is calculated, it can be determined whether series converges or diverges. So, finding the value

\lim_{n \to \infty} (1 -  ( \frac{1}{2} )^{n}) \\  \\  \lim_{n \to \infty} (1 -  \frac{1}{ 2^{n} }) \\  \\ 1 - ( \lim_{n \to \infty} \frac{1}{ 2^{n} }) \\  \\ 1 -  \frac{1}{ 2^{\infty} } \\  \\ 1 -  \frac{1}{\infty} \\  \\ 1 - 0 = 1

Thus
Series Converges to 1
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