Solved on Dec 23, 2023

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

STEP 1

Assumptions
1. We are given a power series of the form: k=1xkk\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(1x)\ln(1-x), which is given by: ln(1x)=k=1xkk\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(1x)\ln(1-x), but with a different sign. We can write: ln(1x)=k=1xkk-\ln(1-x) = \sum_{k=1}^{\infty} \frac{x^{k}}{k}

STEP 4

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

STEP 5

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

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