Question

1. (a) Use the integral test to show that P[infinity] n=1 1/n4 converges. (b) Find the 10th partial sum, s10, of the series P[infinity] n=1 1/n4 . (c) According to the Remainder Estimate for the Integral Test, we know that Z [infinity] n+1 1 x 4 dx ≤ s − sn ≤ Z [infinity] n 1 x 4 dx, (1) where s is the sum of P[infinity] n=1 1/n4 and sn is the nth partial sum of P[infinity] n=1 1/n4 . Use inequality (1) and s10 from

Answer 1

Answer:

Step-by-step explanation:

a) $\int\limits^{\infty} _1 {\frac{1}{n^4} } \, dn\\ =\frac{n^{-3} }{-3}$

Substitute limits to get

= $\frac{1}{3}$

Thus converges.

b) 10th partial sum =

$\int\limits^{10} _1 {\frac{1}{n^4} } \, dn\\ =\frac{n^{-3} }{-3}$

=$\frac{-1}{3} (0.001-1)\\= 0.333$

c) Z [infinity] n+1 1 /x ^4 dx ≤ s − sn ≤ Z [infinity] n 1 /x^ 4 dx, (1)

where s is the sum of P[infinity] n=1 1/n4 and sn is the nth partial sum of P[infinity] n=1 1/n4 .

(question is not clear)