# If cos x is replaced by 1 - (x2/2) and |x |<0.5, what estimate can be made of the error? Does 1 - (x2/2) tend to be too large, or too small? Give reasons for your answer.

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If cos x is replaced by 1 - (x2/2) and |x |<0.5, what estimate can be made of the error? Does 1 - (x2/2) tend to be too large, or too small? Give reasons for your answer.

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Step 1

We’ll answer the second question first. Notice the question have used first two digits of Maclaurin series for cosx. This is an alternating series, irrespective of whether x is positive or negative.

Step 2

For small x the factorials in the denominator will dominate the powers of x in the numerator, so the terms will definitely decrease in magnitude. So it will tend to 0, since the cosine series converges for every x. Thus the conditions for the Alternating Series Test and Alternatin...

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