Question

Consider the convergent alternating series Summation from k equals 1 to infinity StartFraction (negative 1 )Superscript k Over (2 k plus 1 )cubed EndFraction . If the third partial sum Upper S 3 is used to estimate the value of the​ series, what is the upper bound given by the Alternating Series Remainder Theorem on the absolute error |R 3 | in this​ estimation?

Answer 1

Answer:

Step-by-step explanation:

Check attachment for solution.

Let assume n = k, it does not change anything

Given the series

Σ(-1)ⁿ / (2n+1)ⁿ from n=1 to n= infinity

We can express the series as the sum of partial sums and infinite remainder

S = Sn + Rn

The partial sum is already set to the 3rd terms since the upper bound is given to be 3

So, the error should be from the 4th term to infinity.

NOTE: the error is dominated by the first term of the error. The first term of the error is the fourth term.

Therefore,

The error bound = the value of the fourth term

So, when n = 4

Rn = (-1)ⁿ / (2n + 1)³

R4 = (-1)⁴ / (2•4 + 1)³

R4 = 1 / (8 + 1)³

R4 = 1 / 9³

Rn = 1 / 729

The error bound is positive