Final answer:
To approximate ln(1.1) using Taylor's expansion with accuracy of 10⁻⁵, one must sum the series ln(1+x) = x - x²/2 + x³/3 - ... until the absolute value of the last term added is less than 10⁻⁵.
Step-by-step explanation:
The question asks to use Taylor's expansion to compute an approximation of ln(1.1) with accuracy 10⁻⁵. To find how many terms we need, we can use the Taylor series expansion of f(x) = ln(1 + x) around x = 0:
f(x) = f(0) + f'(0)x + f''(0)x²/2! + ... + f(n)(0)xn/n! + ...
Since f(x) = ln(1 + x), we have:
f'(x) = 1/(1 + x), f''(x) = -1/(1 + x)2, f'''(x) = 2/(1 + x)3, ...
The derivatives at x = 0 are:
f'(0) = 1, f''(0) = -1, f'''(0) = 2, ...
The pattern of derivatives is an alternating series of n!/xn+1 for n ≥ 1, thus the Taylor series is:
ln(1 + x) = x - x²/2 + x³/3 - x´/4 + ...
To approximate ln(1.1) with an accuracy of 10⁻⁵, we incrementally sum terms until the absolute value of the last term is less than 10⁻⁵. Each term is (0.1)n/n.