Generate common power series terms and evaluate polynomial approximations
Result
Series Terms
Power Series Rules
1 / (1 - x) = 1 + x + x^2 + x^3 + ... for |x| < 1
e^x = 1 + x + x^2/2! + x^3/3! + ...
sin(x) = x - x^3/3! + x^5/5! - ...
cos(x) = 1 - x^2/2! + x^4/4! - ...
ln(1+x) = x - x^2/2 + x^3/3 - ... for -1 < x ≤ 1
A power series converts a function into a polynomial-like expression. This is useful for approximation, calculus, differential equations, and numerical methods when exact evaluation is difficult.
⚠Always check the interval of convergence. A series can look valid algebraically but fail to represent the function outside its convergence range.
What Is a Power Series?
A power series has the form c0 + c1x + c2x^2 + c3x^3 + ... or c0 + c1(x-a) + c2(x-a)^2 + ... . Near the center, a finite number of terms can approximate many smooth functions.
Approximation
Use the first n terms to build a polynomial approximation that is easy to evaluate and differentiate.
Convergence
The interval of convergence tells where the infinite series matches the original function.
Maclaurin Series
A Maclaurin series is a power series centered at zero, common for sin, cos, exp, and ln functions.
Error Control
Adding terms usually reduces error near the center, but far-away inputs may require many terms or may not converge.
Teaching Example: for e^x at x = 0.5, six terms give 1 + 0.5 + 0.5^2/2! + 0.5^3/3! + 0.5^4/4! + 0.5^5/5!, a close polynomial approximation to e^0.5.
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