Question

Prove that $$ \ln (1+x)=\sum_{k=1}^{n}(-1)^{k+1} \frac{x^{k}}{k}+R_{n}(x), $$ where \(\lim _{n \rightarrow \infty} R_{n}(x)=0\) if \(-1

Short answer

The expression is proven as the limit of the remainder \( R_n(x) \) goes to zero for \(-1<x \leq 1\).

Step-by-step solution

Understand the Taylor Series for ln(1+x)

The Taylor series expansion of the natural logarithm function, \( \ln(1+x) \), can be defined as an infinite series:\[ \ln(1+x) = \sum_{k=1}^{\infty} (-1)^{k+1} \frac{x^k}{k} \]This series converges for \(-1 < x \leq 1\). The series represents the function perfectly as \( n \) approaches infinity.

Define the Remainder Term R_n(x)

When truncating a series at \( n \) terms, the remainder \( R_n(x) \) is the difference between the actual value of \( \ln(1+x) \) and the \( n \)th partial sum:\[ R_n(x) = \ln(1+x) - \sum_{k=1}^{n}(-1)^{k+1} \frac{x^k}{k} \]

Analyze the Limit of the Remainder

The goal is to demonstrate that \( \lim _{n \rightarrow \infty} R_{n}(x) = 0 \).For any \(-1 < x \leq 1\), the series converges to \( \ln(1+x) \), meaning as \( n \) approaches infinity, the remainder \( R_n(x) \) approaches zero because the finite sum approximates the true value of the function better and better.

Conclude with Limit of the Remainder

Since for \(-1 < x \leq 1\), the series \( \sum_{k=1}^{\infty} (-1)^{k+1} \frac{x^k}{k} \) converges to \( \ln(1+x) \), and the definition of convergence implies that the remainder \( R_n(x) \) must go to zero:\[ \lim_{n \rightarrow \infty} R_n(x) = 0 \]. Hence, the expression is proven.

Key concepts

Natural Logarithm

The natural logarithm, noted as \( \ln(1+x) \), is a fundamental mathematical function. It's defined for positive arguments and particularly insightful within the context of calculus and series.

To gain a deeper understanding, consider how \( \ln(1+x) \) can be expressed using a Taylor series. A Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a single point. This approach allows us to approximate functions that may not be easily integrated or differentiated otherwise.

For the natural logarithm function, the series representation is:
\[ \ln(1+x) = \sum_{k=1}^{\infty} (-1)^{k+1} \frac{x^k}{k} \] This series effectively gives us a powerful tool for calculating the natural logarithm for any value of \( x \) in the interval \(-1 < x \leq 1 \).

By using this series, we can compute logarithms without a calculator, simply by adding up a sequence of terms. The alternating sign of the terms (\((-1)^k\)) ensures that each term corrects the previous one, which helps to improve the approximation as more terms are added.

Series Convergence

Series convergence is an essential concept in calculus, especially when dealing with infinite series representation like the Taylor series. It refers to whether the sum of an infinite sequence of numbers approaches a finite limit as more terms are added.

When we say the Taylor series for \( \ln(1+x) \) converges, we mean that as we include more terms in the series \( \sum_{k=1}^{\infty} (-1)^{k+1} \frac{x^k}{k} \), the series gets closer and closer to the actual value of the natural logarithm for \( x \).
For the specific series of the natural logarithm, the series converges in the domain \(-1 < x \leq 1\). This convergence implies that by increasing \( n \), the approximations become more accurate, eventually matching the true value of \( \ln(1+x) \) perfectly in theory.

Convergence is a powerful property because it reassures us that the approximation will eventually reach the desired accuracy. It ensures the series representation is valid and reliable within the specified bounds, offering a rigorous method for approximating \( \ln(1+x) \).

Remainder Term

The remainder term, \( R_n(x) \), is a crucial part of series approximations. It accounts for the error between the partial sum of the series and the actual value of the function being approximated.

In the context of \( \ln(1+x) \), after truncating the infinite series at \( n \) terms, \( R_n(x) \) is defined by the formula:
\[ R_n(x) = \ln(1+x) - \sum_{k=1}^{n}(-1)^{k+1} \frac{x^k}{k} \]
This remainder tells us how much error there is in using an \( n \)-term sum to estimate \( \ln(1+x) \).

A remarkable property of this series is that if the series is convergent, the remainder \( R_n(x) \) approaches zero as \( n \) approaches infinity. In practical terms, this means that by choosing sufficiently large \( n \), the error \( R_n(x) \) can be made arbitrarily small within the convergence interval \(-1 < x \leq 1 \).

Thus, understanding \( R_n(x) \) helps not just in knowing how good an approximation is, but also in determining how many terms are needed to achieve a desired level of accuracy. This knowledge allows mathematicians and engineers to use series effectively in calculations without having infinitely many terms.