All Labs

Series & Taylor Approximation Lab

Investigate infinite series convergence, build Taylor polynomials term by term, and find intervals of convergence for power series. Apply six convergence tests, watch polynomials approach functions as the degree increases, and explore how differentiation and integration affect power series.

Guided Experiment: Taylor Approximation Accuracy

How does the degree of a Taylor polynomial affect the accuracy of the approximation? What happens to the error at x = 1 as you increase the degree for sin(x)?

Write your hypothesis in the Lab Report panel, then click Next.

Controls

Number of terms (N)20

Results

∑n=1∞(12)n\sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n
Partial Sum S_20
0.999999
Exact Sum
1.000000
Error: |S_20 - S| = 9.537e-7

Convergence Tests

Ratio Testconverges
L=lim⁡n→∞∣an+1an∣≈0.500000L = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| \approx 0.500000

L = 0.5000 < 1, so the series converges absolutely by the Ratio Test.

Root Testconverges
L=lim⁡n→∞∣an∣n≈0.500000L = \lim_{n\to\infty} \sqrt[n]{|a_n|} \approx 0.500000

L = 0.5000 < 1, so the series converges absolutely by the Root Test.

p-Series Testinconclusive
Not a standard p-series form\text{Not a standard p-series form}

This series is not in the standard p-series form 1/n^p, so the p-series test does not apply.

Alternating Series Testinconclusive
Not an alternating series\text{Not an alternating series}

This test only applies to alternating series of the form (-1)^n b_n.

Comparison Testconverges
0≤12n≤12n(geometric with ∣r∣<1)0 \le \frac{1}{2^n} \le \frac{1}{2^n} \quad (\text{geometric with } |r| < 1)

This is a geometric series with |r| = 0.5 < 1, which converges to r/(1-r) = 1.

Integral Testinconclusive
Integral not easily evaluated for this series\text{Integral not easily evaluated for this series}

The Integral Test requires f(x) to be positive, continuous, and decreasing. Automatic evaluation was not performed for this series.

Partial Sums

Data Table

(0 rows)
#TrialSeries/FunctionDegree/NPartial Sum/P_n(x)Exact ValueErrorRadius RConvergence
0 / 500
0 / 500
0 / 500

Reference Guide

Convergence Tests

Determine whether an infinite series converges or diverges using these standard tests.

Ratio Test: L=lim⁡n→∞∣an+1an∣\textbf{Ratio Test: } L = \lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|

If L < 1 the series converges absolutely. If L > 1 it diverges. If L = 1 the test is inconclusive.

p-Series: ∑1np converges iff p>1\textbf{p-Series: } \sum \frac{1}{n^p} \text{ converges iff } p > 1

Taylor & Maclaurin Series

Approximate a function near a point using polynomial terms built from its derivatives.

f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

A Maclaurin series is a Taylor series centered at a = 0. The more terms you include, the better the approximation near the center point.

Lagrange Error Bound

The Lagrange remainder gives an upper bound on the error of a Taylor polynomial of degree n.

∣Rn(x)∣≤M⋅∣x−a∣n+1(n+1)!|R_n(x)| \le \frac{M \cdot |x - a|^{n+1}}{(n+1)!}

Where M is the maximum value of |f^(n+1)(c)| for c between a and x. This bound is essential for determining how many terms are needed for a desired accuracy.

Interval of Convergence

Every power series has a radius R such that the series converges for |x - a| < R and diverges for |x - a| > R.

R=lim⁡n→∞∣anan+1∣R = \lim_{n\to\infty} \left|\frac{a_n}{a_{n+1}}\right|

Endpoints must be checked separately, as the series may converge or diverge at x = a - R and x = a + R. Differentiating or integrating a power series preserves the radius but may change endpoint behavior.