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Power Series Calculator

Expand eˣ, sin, cos, ln, √x, arctan and six more about any centre a, with exact fractional coefficients, the general term, and the interval of convergence including both endpoints.

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Power Series Calculator works straight from this page — free, instant, nothing to install. Its place is under Sequences & series; Infinite Series Sum Calculator and Series Convergence Test Calculator answer the questions closest to this one.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is Power Series Calculator?

Expand eˣ, sin, cos, ln, √x, arctan and six more about any centre a, with exact fractional coefficients, the general term, and the interval of convergence including both endpoints.

When would I actually use this?

Finding the nth term without listing every one before it, summing a long run in one step, and recognising which family a sequence belongs to.

What is the most common mistake?

Starting the index at the wrong end. Whether the first term is a₀ or a₁ shifts every result by one position, and the two conventions are both common.

How is Power Series Calculator different from Infinite Series Sum Calculator?

They sit next to each other but answer different questions. Infinite Series Sum Calculator: ten classic families — geometric, p-series, telescoping, alternating harmonic, Leibniz, Basel, Σ1/n! — with their closed form (π²/6, ln 2, π/4, e), the convergence condition and the partial sums. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Series Convergence Test Calculator is the closest one after this: Pick a series and see which test settles it — p-series, ratio, root, Leibniz, Cauchy condensation, limit comparison — with the limit it produces and whether the convergence is absolute or conditional.

What else is worth having open alongside it?

Fourier Series Coefficients Calculator and Maclaurin Series Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The closed forms are the standard ones, and the tool states the indexing convention it uses so a result can be compared against a textbook without ambiguity.

Further reading

All guides →
ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.ExplainerSprings in Series Behave Like Resistors in ParallelTwo 500 N/m springs end to end give 250 N/m; side by side they give 1,000. Under the same 50 N load that is 0.2 m of travel against 0.05 m, and 5 joules stored against 1.25.ExplainerCapacitors in Series: the Smallest One Decides, and Then It Fails100, 220 and 330 µF in series come to 56.9 µF — less than the smallest. Put 100 µF and 1 µF in series and the total is 0.99 µF, while the small one carries 99 % of the voltage.ExplainerResistors in Series and in Parallel: The Formulas, and the Three Things They Do Not Tell YouSeries adds, parallel adds reciprocals — that part is easy. The parts that catch people: which member dominates the answer, why two 5 % resistors in parallel are still 5 %, and why the smallest resistor in a parallel pair takes the largest share of the power.How-toHow to Calculate an LED Series ResistorSize the series resistor for any LED with one formula: R = (Vs − Vf) / I. Follow the steps, see a worked 5 V example, and pick the right resistor.ExplainerA 5,000 mAh Power Bank Does Not Hold 5,000 mAh for Your Phone5,000 mAh drawn at 500 mA with 85 % efficiency runs for 8.5 hours instead of the arithmetic's 10. The same 15 % explains why a 5,000 mAh bank barely fills a 4,000 mAh phone once.