How do partial sums approximate the function defined by the series $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots$?

Conditions

• The approximation is evaluated near the center of expansion ($x=0$).
• A finite number of terms is selected from the infinite series.

Reasoning, step by step

1. Select the first $n$ terms of the series to form a partial sum polynomial.
2. Plot the polynomial on a coordinate plane alongside the target function.
3. Observe that the polynomial closely matches the target function near $x=0$.
4. Add more terms to the partial sum and observe that the region of close agreement expands outward.
5. Conclude that finite polynomials are local approximations that improve with more terms.

Example

The video demonstrates plotting the first two terms ($1 - \frac{x^2}{2!}$), which gives a downward parabola, and then adding the third term ($+ \frac{x^4}{4!}$), which creates a 'W' shape that hugs the actual function curve much longer.

Common misconceptions

• Assuming a finite partial sum will perfectly match the target function everywhere.
• Believing that adding terms only improves the approximation at exactly $x=0$ without expanding the valid interval.

Watch the explanation

Explore next

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• How do individual terms affect the shape of the approximation?