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Infinite Seriesmedium
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What is the smallest integer NNN such that the remainder RNR_NRN​ of the series ∑n=1∞(−1)nn4\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4}∑n=1∞​n4(−1)n​ satisfies ∣RN∣<0.001|R_N| < 0.001∣RN​∣<0.001?