Question

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.ln (1 + x²)

Accepted Answer

Recall the Taylor series expansion for $$ \ln(1+x) $$ centered at 0, which is given by:

$$ 
\ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots 
$$  
This is the series you will manipulate to find the series for $$ \ln(1+x^2) . $$Substitute $$ x^2  in $$place of $$ x  in $$the series for $$ \ln(1+x) . $$This gives:

$$ 
\ln(1+x^2) = x^2 - \frac{(x^2)^2}{2} + \frac{(x^2)^3}{3} - \frac{(x^2)^4}{4} + \cdots 
$$  
which simplifies to:

$$ 
\ln(1+x^2) = x^2 - \frac{x^4}{2} + \frac{x^6}{3} - \frac{x^8}{4} + \cdots 
$$ Identify the first four nonzero terms from the series after substitution. These terms correspond to the powers of $$ x $$ with nonzero coefficients in the expansion. Write out explicitly the first four nonzero terms of the Taylor series for $$ \ln(1+x^2) $$ centered at 0, using the simplified powers and coefficients from the previous step. Verify that the series is centered at 0 and that the terms are ordered by increasing powers of $$ x . $$This confirms the correctness of the Taylor series expansion for $$ \ln(1+x^2) .$$

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

ln (1 + x²)

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Key Concepts

Taylor Series Expansion

Manipulating Known Series

Identifying Nonzero Terms