Representing Functions as Power Series, Practice Problems, Examples - Calculus

Functions can be represented by power series by manipulating them to fit the form of a geometric series, which converges to \(a/(1-r)\) for \(|r| lt 1\). By replacing '\(r\)' with a function of '\(x\)' and '\(a\)' with a constant, a power series representation for a function can be found by plugging the expression for '\(r\)' into the geometric series formula, such as \(1/(1-x)=\sum _{n=0}^{\infty }x^{n}\).

💡Method 1: Manipulating the geometric series
• Start with the basic formula: The geometric series formula is \(1/(1-x)=\sum _{n=0}^{\infty }x^{n}\) for \(|x| lt 1\).
• Manipulate the function: Rewrite your function to have the form \(a/(1-r)\), where '\(a\)' is a constant and '\(r\)' is an expression involving '\(x\)'.
• Example: To find the power series for \(1/(1+x)\), you can rewrite it as \(1/(1-(-x))\). Here, \(a=1\) and \(r=-x\).
• Substitute into the series: Substitute the expression for '\(r\)' into the geometric series formula to find the power series representation.
For \(1/(1+x)\), the series is \(\sum _{n=0}^{\infty }(-x)^{n}=\sum _{n=0}^{\infty }(-1)^{n}x^{n}\), which is \(1-x+x^{2}-x^{3}+\dots \).

💡Method 2: Using partial fractions
• Decompose the function: If the function is a rational function, use partial fraction decomposition to break it into simpler terms, as shown in this YouTube video.
• Represent each term: Find the power series for each individual fraction using the geometric series method.
• Combine the series: Add the power series for each term together to find the series for the original function.
• Determine the interval of convergence: The interval of convergence for the entire function is the intersection of the intervals of convergence for each of the individual series.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1DlYi7xcmyEAhWTnwUN8zRgisBdHIdmtr/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1VEHFVDs67vvvQVVsozjmOCO5RbEgzbkA/view?usp=drive_link

💡Chapters:
00:00 Functions as power series
01:14 Worked examples

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