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math104-s21:hw6 [2021/02/26 21:29] pzhou created |
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The second theme is series. It is related to sequence by considering partial sum, and we reuses many of results from sequence. The alternating series test and the integral test was new, but one should have met them in calculus. | The second theme is series. It is related to sequence by considering partial sum, and we reuses many of results from sequence. The alternating series test and the integral test was new, but one should have met them in calculus. | ||
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1. Rudin Ex 2.11. | 1. Rudin Ex 2.11. | ||
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6. If $a_n > 0$ and $\sum_n a_n$ converges, show that $\sum_n \sqrt{a_n}/ | 6. If $a_n > 0$ and $\sum_n a_n$ converges, show that $\sum_n \sqrt{a_n}/ | ||
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+ | Hint: Cauchy inequality: for any real valued $N$-tuples $(A_1, \cdots, A_N)$ and $(B_1, \cdots, B_N)$, we have $$ (\sum_{j=1}^N A_j B_j)^2 \leq (\sum_{j=1}^N A_j^2) (\sum_{j=1}^N B_j^2)$$ | ||