Question

find the limit for the sequence

Answer 1

\[\sqrt{\left( 1+\frac{ 1 }{ 2n } \right)}^{n}\]

Answer 2

can someone show me how to solve the problem thank you

Answer 3

hint:
\[\Large \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e\]

Answer 4

can you show me how? i really don't get this section

Answer 5

\[\Large \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e\]
\[\Large \lim_{2n\to\infty}\left(1+\frac{1}{2n}\right)^{2n} = e\]
\[\Large \left(\lim_{2n\to\infty}\left(1+\frac{1}{2n}\right)^{2n}\right)^{1/4} = \left(e\right)^{1/4}\]
\[\Large \lim_{2n\to\infty}\left(\left(1+\frac{1}{2n}\right)^{2n}\right)^{1/4} = \left(e\right)^{1/4}\]
\[\Large \lim_{2n\to\infty}\left(1+\frac{1}{2n}\right)^{2n*1/4} = e^{1/4}\]
\[\Large \lim_{2n\to\infty}\left(1+\frac{1}{2n}\right)^{n/2} = e^{1/4}\]
\[\Large \lim_{2n\to\infty}\left(\sqrt{1+\frac{1}{2n}}\right)^n = e^{1/4}\]
\[\Large \lim_{n\to\infty}\left(\sqrt{1+\frac{1}{2n}}\right)^n = e^{1/4}\]

Answer 6

It might help to think in reverse. Start with what you want to find, then work backwards. That's one way to do it