Fractions and Decimals

Discrete Mathematics

Differential Equations

Diagram & Picture

Numbers & Operations

Data & Statistics

Natural Numbers

Ratios & Proportions

Order of Operations

Exponents & Radicals

Complex Numbers

Congruence & Similarity

Pythagorean Theorem

Transformations

Data Collection & Representation

Measures of Central Tendency

Measures of Spread

Statistical Inference

Non-Right Triangle

Unit Circle & Radians

Identities & Equations

Limits & Continuity

Sequences & Series

Solved on Dec 23, 2023

Identify the function represented by the power series $\sum_{k=1}^{\infty} \frac{x^{k}}{k}$ . Find the corresponding Taylor series $f(x)$ .

STEP 1

Assumptions
1. We are given a power series of the form: $\sum_{k=1}^{\infty} \frac{x^{k}}{k}$
2. We need to identify a common function that this power series represents.
3. We will compare the given power series to known Taylor series expansions of common functions.

STEP 2

Recall the Taylor series expansion for the natural logarithm function, $\ln(1-x)$ , which is given by: $\ln(1-x) = -\sum_{k=1}^{\infty} \frac{x^{k}}{k}$

STEP 3

Notice that the given power series has a similar form to the Taylor series of $\ln(1-x)$ , but with a different sign. We can write: $-\ln(1-x) = \sum_{k=1}^{\infty} \frac{x^{k}}{k}$

STEP 4

Since we are looking for a function $f(x)$ such that: $f(x) = \sum_{k=1}^{\infty} \frac{x^{k}}{k}$ we can conclude that: $f(x) = -\ln(1-x)$

STEP 5

However, the function $\ln(1-x)$ is valid for $-1 < x < 1$ . Therefore, the function $f(x)$ represented by the given power series is: $f(x) = -\ln(1-x)$ for $-1 < x < 1$ .
The function represented by the given power series is $f(x) = -\ln(1-x)$ .

Was this helpful?

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Contact Influencer program Policy Terms