- #1

KungPeng Zhou

- 22

- 7

- Homework Statement
- ##a_{n}=\int_{0}^{2-\sqrt{3}}\frac{1-x^{4n}}{1+x^{2}}dx##, evaluate ##\underset{n\rightarrow \infty} {\lim}a_{n}##

- Relevant Equations
- FTC

In my opinion , if it can be shown that this is a monotonically bounded sequence, one can confirm that there is a limit.

First，we know $$ \frac{1-x^{4n}}{1+x^{2}}dx=(1-x^{2}) (1+x^{2}) ^{n-1}=(1-x^{4}) ^{n-1}(1+x^{2}).$$

According to the integral median theorem，we can get $$a_n=(2- \sqrt{3} ) (1-\alpha^{4})^{n-1}(1+x^{2}), \alpha\in[0,2-\sqrt{3}]$$

But I don't know how to continue with the question.

First，we know $$ \frac{1-x^{4n}}{1+x^{2}}dx=(1-x^{2}) (1+x^{2}) ^{n-1}=(1-x^{4}) ^{n-1}(1+x^{2}).$$

According to the integral median theorem，we can get $$a_n=(2- \sqrt{3} ) (1-\alpha^{4})^{n-1}(1+x^{2}), \alpha\in[0,2-\sqrt{3}]$$

But I don't know how to continue with the question.

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