ind the Taylor series expansion of f(x) = e^(x²) centered at x=0 and determine its interval of convergence.
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Step-by-Step Solution
Step 1: Recall the Maclaurin series for eu
The Maclaurin series is a special case of the Taylor series where the expansion is centered at x=0. We know the Maclaurin series for eu is given by the sum from n=0 to infinity of un divided by n!. This series converges for all real values of u.
Step 2: Substitute u=x2 into the series
To find the Taylor series for f(x)=ex2 centered at x=0, we substitute u=x2 into the Maclaurin series for eu. This gives us the series in terms of x.
Step 3: Write out the first few terms of the series
Expanding the series, we can write out the first few terms to better visualize the pattern. This gives us 1+x2+2x4!+3x6!+….
Step 4: Determine the interval of convergence
We use the Ratio Test to find the interval of convergence. Let an=nx2n!. We calculate the limit of the absolute value of the ratio of consecutive terms, anan+1. Since the limit is 0, which is less than 1, the series converges for all real values of x.