Question

enor that is made by approximoting the I k: k5 1.a) Eslimote the the sum of the given Series by the sum of 1.b) for the Series given, delermine how lorgen must be first five terms So that the Series gives Slated using the 1th partial sum to approximate on error of no more than the error o t t4 : En < 0.00003. kal

Answer 1

Part (a) : First we will calculate the sum of first 10 terms.

10 terms = 1+1/2⁵+1/3⁵+1/4⁵+1/5⁵= 1.036667

Now we will solve for error using integral approximation : -

Error (E) =  $\int_{10}^{ \infty } \frac{1}{x{5}}dx$ = 2.5*$10^{-5}$ (approx).

Part (b) :

The series above satisfies all three conditions of the alternating series test. Using the inequality above, we need to find an 'n' such that:

∣s−s_n∣ ≤ ∣a_n+1∣  $=\left | \frac{1}{(n+1)^{4}} \right |< 0.00003$

Now => (n+1)⁴ > 33333.333

n > 12.51

From this we can conclude that n > 12.51 for error less then 0.00003.