In our calculus textbook, Thomas' Calculus, in Chapter 10.8, we are asked to find the first three nonzero terms of the Taylor approximation of the function $f(x) = \sin(x)\cdot\ln(1+x)$ as well as find its interval of convergence.
In order to do this, I multiplied out the Taylor polynomials: $$(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...) \cdot (x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+...) $$ Then, I distributed the terms of the $\ln(1+x^2)$: $$x(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...)-\frac{x^2}{2}(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...) + \frac{x^3}{3}(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...) -\ldots $$
Because the problem asked for the first three nonzero terms, I noticed that the first three nonzero terms will be of degree $1,2,$ and $3$. Grouping and distributing these terms, I got the final answer $\boxed{{x^2 - \frac{x^3}{2} + \frac{x^4}{6}}}.$ And, since the interval of convergence of $\ln(1+x)$ is $(-1,1]$, the answer only converges for $-1<x\leq1 $.
However, others' methodologies involved multiplying their Maclaurin series representations together and disregarding the summation signs. $$\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}\cdot \sum_{n=0}^\infty \frac{(-1)^{n}x^{n+1}}{n+1}.$$
The second series is obtained from shifting the index of the Maclaurin series of $\ln(1+x)$. The result they achieved was: $$\sum_{n=0}^\infty \frac{x^{3n+2}}{(2n+1)!(n+1)} .$$
The teacher agreed with this methodology; however, I had previously thought that one cannot multiply two series and then sum the product of the functions as it is not a property of sums to do so. Furthermore, plugging this series in to WolframAlpha I obtained that this series represents $\frac{2\cosh((x^{3/2}) - 1)}{x}$. And determining the interval of convergence was left as a further exercise outside of class.
Therefore, I have two questions:
- Which method was correct in determining and answering the problem?
- Are there any conditions or situations in which one can multiply two power series in order to determine the Taylor representation of the product of two functions?
Any help would be appreciated.
Your product is incorrect.
$$\sum_{i=0}^{\infty}a_ix^i.\sum_{j=0}^{\infty}b_jx^j=$$ $$\sum_{i,j\ge 0}a_ib_jx^{i+j}=$$
$$\sum_{n=0}^{\infty}\Bigl (\sum_{i=0}^na_ib_{n-i}\Bigr)x^n $$