Question

\({\sum\limits_{k = 1}^\infty {\left( {\cos (1)} \right)} ^k}\) Find Whether It Is Convergent Or Divergent. If It Is Convergent Find Its Sum.

Short answer

Given \(\begin{aligned}{\sum\limits_{k = 1}^\infty {\left( {\cos (1)} \right)} ^k}\\&= \cos (1) + {\left( {\cos (1)} \right)^2} + {\left( {\cos (1)} \right)^3} + .........\end{aligned}\)

The \({\sum\limits_{k = 1}^\infty {\left( {\cos (1)} \right)} ^k}\) is a Geometric Series

Step-by-step solution

Find The Limit Of \({\left( {\cos (1)} \right)^k}\) As \(k \to \infty \)

Consider \(\mathop {\lim }\limits_{k \to \infty } {\left( {\cos (1)} \right)^k} = \mathop {\lim }\limits_{k \to \infty } {\left( {\cos (1)} \right)^k}\)

The Series \({\sum\limits_{k = 1}^\infty {\left( {\cos (1)} \right)} ^k}\) is a Geometric Series with \(a = \cos (1)\) and \(r = \cos (1)\) ,where a is the first term of the Geometric Series and r is the Common Ratio then the limit is \(\frac{a}{{1 - r}}\) if \(\left| r \right| < 1\) ;

Here \(r = \cos (1) = 0.99984769\) i.e \(\left| {\cos (1)} \right| < 1.\) Hence \({\sum\limits_{k = 1}^\infty {\left( {\cos (1)} \right)} ^k}\) Converges

\( \Rightarrow \mathop {\lim }\limits_{k \to \infty } {\left( {\cos (1)} \right)^k} = \frac{{\cos (1)}}{{1 - \cos (1)}}\)

\( \approx 1.175343\)

Finding The Summation

\({\sum\limits_{k = 1}^\infty {\left( {\cos (1)} \right)} ^k} = \mathop {\lim }\limits_{k \to \infty } {\left( {\cos (1)} \right)^k}\)

\( = 1.17543\)

Hence\({\sum\limits_{k = 1}^\infty {\left( {\cos (1)} \right)} ^k}\) Convergesand its Sum is\({\sum\limits_{k = 1}^\infty {\left( {\cos (1)} \right)} ^k} = 1.17543\)