This is a Test of Mathematics Solution Subjective 155 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta
Evaluate: $ \lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+n})$
As the title suggests the modification of this problem will be, that we will solve a more general series and then use a specific value to arrive at the solution of this problem.
First let us consider the following limit:
$ \lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+kn})$
Observe carefully that using k=1 in this limit, we get the limit that has been asked to evaluate.
Now
$ \lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+n}) = \lim_{n\to\infty} (\sum_{r=1}^{kn} \frac{1}{n+r})$
$ = \lim_{n\to\infty} (\sum_{r=1}^{kn} \frac{\frac{1}{n}}{1+\frac{r}{n}})$
$ = \lim_{n\to\infty}\frac{1}{n} (\sum_{r=1}^{kn} \frac{1}{1+\frac{r}{n}})$
Let's substitute $ \frac{r}{n} = x => dr = ndx$
Now we can change the sum to an integral
$ => \lim_{n\to\infty} (\sum_{r=1}^{kn} \frac{\frac{1}{n}}{1+\frac{r}{n}}) = \lim_{n\to\infty} \frac{1}{n}*n\int_{0}^{k} \frac{1}{1+x} dx $
$ = \lim_{n\to\infty}( log |x+1|_{k} - log |x+1|_{0})$
$ = \lim_{n\to\infty} log |k+1|$
$ = log |k+1|$ (As the term is an 'n' free term)
So we see the solution is $ = log |k+1|$
Substituting k=1, we get
$ \lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+n}) = \log {2}$
This is a Test of Mathematics Solution Subjective 155 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta
Evaluate: $ \lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+n})$
As the title suggests the modification of this problem will be, that we will solve a more general series and then use a specific value to arrive at the solution of this problem.
First let us consider the following limit:
$ \lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+kn})$
Observe carefully that using k=1 in this limit, we get the limit that has been asked to evaluate.
Now
$ \lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+n}) = \lim_{n\to\infty} (\sum_{r=1}^{kn} \frac{1}{n+r})$
$ = \lim_{n\to\infty} (\sum_{r=1}^{kn} \frac{\frac{1}{n}}{1+\frac{r}{n}})$
$ = \lim_{n\to\infty}\frac{1}{n} (\sum_{r=1}^{kn} \frac{1}{1+\frac{r}{n}})$
Let's substitute $ \frac{r}{n} = x => dr = ndx$
Now we can change the sum to an integral
$ => \lim_{n\to\infty} (\sum_{r=1}^{kn} \frac{\frac{1}{n}}{1+\frac{r}{n}}) = \lim_{n\to\infty} \frac{1}{n}*n\int_{0}^{k} \frac{1}{1+x} dx $
$ = \lim_{n\to\infty}( log |x+1|_{k} - log |x+1|_{0})$
$ = \lim_{n\to\infty} log |k+1|$
$ = log |k+1|$ (As the term is an 'n' free term)
So we see the solution is $ = log |k+1|$
Substituting k=1, we get
$ \lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+n}) = \log {2}$