- #1
ssh
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Show that,
\[\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+\cdots\]
\[\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+\cdots\]
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ssh said:Show that log(1+x) = x - x2\2 + x3\3...
ssh said:Can we write this as a Taylor's series as f(x) = Log(1+x), then f'(x)=1\1+x so on.
ssh said:Show that,
\[\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+\cdots\]
Saknussemm said:Taking the derivative of the MacLaurin series gives you
$1 -x +x^2 - x^3 + x^4 + \ldots$
Since this is a geometric series with ratio $-x$, it equals $\frac{1}{1 + x}$ when x is in $(-1, 1)$.
This shows the expression $\ln(1+x)$ and its MacLaurin expansion to have the same derivative over $(-1, 1)$, which means they are equal within a constant. And, since they are equal at $x=0$, this constant is zero.
If my reasoning is correct, this is simpler than proving the limit of the Taylor remainder.
HallsofIvy said:It is, however, the essentially the same as Fernando Revilla's suggestion in the first response to this thread.
A power series solution for Log(1+x) is an infinite series of the form log(1+x) = ∑(n=1 to ∞) (−1)^(n−1) x^n / n. It is a way of representing the natural logarithm function as an infinite sum of powers of x. This series is valid for all values of x in the interval (-1,1) and can be used to approximate log(1+x) for other values of x.
The power series solution for Log(1+x) can be derived using the Taylor series expansion of the natural logarithm function. By repeatedly taking derivatives of log(1+x) and evaluating them at x=0, we can find the coefficients of the power series. This results in the series mentioned in the answer to the previous question.
One advantage of using a power series solution for Log(1+x) is that it allows for the approximation of log(1+x) for values of x outside the interval (-1,1). Additionally, the power series can be truncated at a certain term to obtain a more accurate approximation of the logarithm function. This method is also useful for solving logarithmic equations.
One limitation of using a power series solution for Log(1+x) is that it is only valid for values of x in the interval (-1,1). Outside of this interval, the series may not converge. Additionally, the accuracy of the approximation decreases as x moves further away from 0. This method also requires knowledge of calculus and the Taylor series expansion.
A power series solution for Log(1+x) can be used in various real-world applications, such as in finance and economics, where the natural logarithm function is commonly used. It can also be used in physics and engineering to approximate various equations involving logarithms. Additionally, it can be used in computer programming to calculate logarithmic functions, as many programming languages do not have a built-in logarithm function.