What is an interval, |S − S5| ≤ |R5| for which the actual sum exists?

Question

The series:

_∞

∑ ((-1)ⁿ) • ((x²ⁿ⁺¹) / (2n+1)!

ⁿ⁼⁰

has a partial sum S₅ = 263/315 when x = 1. What is an interval, |S − S5| ≤ |R5| for which the actual sum exists? Provide an exact answer and justify your conclusion.

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Answer 1

When x=1, the error in the 5th partial sum does not exceed 1/7! in absolute value because the error in an alternating terms series does not exceed the first omitted term.

P{lease note that this is the Taylor series for sin x and converges for all values of x.