- #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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