Solved find the sum of the series (1/n+1/(n+1)+1/(n+2)+…..+1/4n as limit n approaches infinity

August 21, 2021Posted inAdvanced Math, Discrete Mathematics, Linear Algebra, Mathematics, Multivariable Calculus, Number Theory, Numerical Analysis, Real Analysis
Question : Find the sum of the series 1/(n+1)+1/(n+2)+…..+1/4n as limit n approaches infinity Solution : Let the sum of the series as \[L=\lim_{n\rightarrow\infty}{\left[\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots..+\frac{1}{4n}\right]\ }\] \[L=\lim_{n\rightarrow\infty}{\left[\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots..+\frac{1}{n+3n}\right]\ }\] \[=\lim_{n\rightarrow\infty}{\frac{1}{n}\left[\frac{n}{n}+\frac{n}{n+1}+\frac{n}{n+2}+\ldots..+\frac{n}{n+3n}\right]\ }\] \[=\lim_{n\rightarrow\infty}{\frac{1}{n}\left[\frac{1}{1+\frac{0}{n}}+\frac{1}{1+\frac{1}{n}}+\frac{1}{1+\frac{2}{n}}+\ldots..+\frac{1}{1+\frac{3n}{n}}\right]\…

Solved Sum of the series 1/1.2-1/2.3+1/3.4-1/4.5+…

August 19, 2021Posted inAbstract Algebra, Advanced Math, Multivariable Calculus, Number Theory, Real Analysis, Trigonometry
Question : Sum of the series 1/1.2-1/2.3+1/3.4-1/4.5+… Solution : Let the Sum of the series is \[S=\ \frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}-\frac{1}{4.5}+\ldots..\] \[=\left(\frac{1}{1}-\frac{1}{2}\right)-\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)-\left(\frac{1}{4}-\frac{1}{5}\right)+\ldots.\] \[=\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots.\right)+\left(-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\ldots..\right)\] \[=\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots.\right)+\left(-1+1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\ldots..\right)\] We shall use formula of the series \[\log{\left(1+x\right)}=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots..\] where…