1. Expand the Sum: Use the sigma notation $\sum c_n x^n$ to write out the first few terms.
2. Identify the Coefficients ($c_n$): These are the "Weights" of each power.
3. Check for Convergence: Does the value of $x$ stay within the allowed range?
4. Sum the Geometric: If it fits the form, use $S = a / (1-r)$.
Write out the first 4 terms of each power series.
$\sum_{n=0}^{\infty} x^n$
$\sum_{n=1}^{\infty} \frac{x^n}{n}$
$\sum_{n=0}^{\infty} (2n)x^n$
Using the rule $\frac{1}{1-x} = \sum x^n$, find the series for each function.
$f(x) = \frac{1}{1 - 2x}$
$f(x) = \frac{1}{1 + x}$
In the series $1 + x + x^2 + x^3...$, what happens if $x = 1$? Calculate the sum of the first 5 terms. Then the first 10. Does it settle on a number? Why is $x=1$ the boundary of the "Interval of Convergence"?
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Use your calculator to see if the series converges for the given $x$.
Series: $1 + x + x^2 + ...$ with $x = 0.9$.
Find the sum of the first 5 terms.
Series: $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ with $x = 2$.
(Note: $n! = n \times (n-1) \times ...$)
Calculate the first 4 terms: $1 + 2 + 2^2/2 + 2^3/6$.
Does it look like it's getting smaller or bigger?
The function $f(x) = e^x$ can be modeled by the power series:
$1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + ...$
Task: If $x = 0.1$, calculate the sum of these first 5 terms. Then calculate $e^{0.1}$ on your calculator. How close was the "Polynomial Approximation"?
Objective: Explain "Modeling" to a younger sibling using a magnifying glass.
The Activity:
1. Draw a circle on a paper.
2. Have them look at a tiny part of the circle with a magnifying glass.
3. Ask: "Does it look like a circle now, or a straight line?" (Straight).
4. "A Power Series is like a very smart magnifying glass that uses straight lines to build the circle."
The Lesson: "God made the world so that we can understand big, round things by looking at small, straight things."
Response: ___________________________________________________________