find the number of terms please help

Question

For the series below calculate find the number of terms  that must be added in order to find the sum to the indicated accuracy.



|error| 

Answer 1

The standard process for this is to state that, as this is an alternating series, the error is less than the absolute value of the next term.

If 2/n^.8 < .01, .01n^.8 > 2, or n^.8 > 200, or (n^.8)^1.25 > 200^1.25, or n > 200^1.25 > 752.12

The 753rd term will have absolute value less than .01, so n = 752 is sufficient.

(Further analysis showed that approximately 316 is sufficient; you can actually look for the first term with absolute value less than twice the error at some point given the smooth nature of the slow decline; 100^1.25 = 316.2278.)

Answer 2

I take it as the series being ∑₁^∞(-1)ⁿ[ 2 /( n^4/5)]

The series is an alternating one with b_n = 2 /( n^4/5)

1. b_n+1 ≤ b_n and
2. lim b_n =0

Therefore by the Alternating Series Estimating Theorem | R_n | = | s − s_n | ≤ b_n+1

which tells us that the size of the error is less than b_n+1.

b₇₅₂ = 0.01001 and b₇₅₃= 0.00991

That is the 753rd term is less than the desired error, we need to add the first 752 terms to get the sum to the desired accuracy.