How to show that $\lim_{n\rightarrow \infty }{a_n}^{b_n}=\alpha ^\beta $?

Question

If $\lim_{n\rightarrow \infty }{a_n}=\alpha (\neq 0) $ and $\lim_{n\rightarrow \infty }{b_n}=\beta$, then $\lim_{n\rightarrow \infty }{a_n}^{b_n}=\alpha ^\beta $?

I unconsciously used this but I realized I'd never seen this theorem before. Is it true?

See also: convergent sequence ${a_n},{b_n}$, does it convergent ${ {a_n}^{b_n} }$? — Martin Sleziak, Dec 23 '19 at 10:17

score 6 · Accepted Answer · edited Jan 02 '20 at 21:00

6

Notice that $$a_n^{b_n}=e^{b_n\log(a_n)}$$ so by the continuity of the exponential and logarithmic functions you have the result, of course with the assumption $\boldsymbol{a_n>0}$ and $\boldsymbol{\alpha>0}$.

edited Jan 02 '20 at 21:00

Daniele Tampieri

9,727

answered Jan 08 '14 at 14:46

score 2 · Answer 2 · answered Jan 08 '14 at 14:43

2

I think that we can use the fact of convergent sequences product and that $$e^{b_n \cdot \log a_n}$$ converges when $n \rightarrow \infty$.

answered Jan 08 '14 at 14:43

Mario De León

1,698

How to show that $\lim_{n\rightarrow \infty }{a_n}^{b_n}=\alpha ^\beta $?

2 Answers2

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