Partial Sum Visualizer

Math Algebra • Sequences and Series

Written by STEM Calculators Team Published December 19, 2025 Updated February 24, 2026

10. Partial Sum Visualizer

Compute and plot partial sums S_n for common series families. Animate summation, detect oscillations, and show sum estimates/bounds.

Series preset: Max n (N): Exponent p (p-series): Geometric first term a: Geometric ratio r:

Animate summing (Play/Pause) Show point labels (sparse) Smooth line (connect points) Show term points (aₙ)

Animation speed (terms/sec):

Ready

Drag to pan • wheel/pinch to zoom • Auto-fit recommended after changing N

Partial sum plot + term points

Top: S_n vs n • Bottom strip: a_n vs n (current term blinks)

n: 0, S: 0 sx: 8, sy: 80 Current: n=0, aₙ=0, Sₙ=0 Sₙ aₙ current (blink)

Choose a series and click Calculate.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Theory: Partial Sums and Term Behavior

A series is built from its terms \(a_n\). The running total is the sequence of partial sums. This tool plots both: top: \(S_n\) vs \(n\), and bottom strip: term points \((n,a_n)\). During animation, the current term point blinks so you can see what is being added.

1) Terms and partial sums

\[ S_n = \sum_{k=1}^{n} a_k \qquad\Longrightarrow\qquad S_n - S_{n-1} = a_n. \]

The series \(\sum a_n\) converges to \(S\) if \(S_n \to S\) as \(n\to\infty\).
A necessary condition for convergence is \(a_n \to 0\). If terms do not approach 0, the series diverges.
Even when \(a_n \to 0\), the series may still diverge (example: harmonic series \(1/n\)).

2) Alternating series: oscillation pattern and error bound

An alternating series has the form:

\[ \sum_{n=1}^{\infty} (-1)^{n+1} b_n, \qquad b_n \ge 0. \]

If \(b_n\) decreases and \(b_n\to 0\), then the series converges and the partial sums typically “zig-zag” toward the limit. A useful truncation bound is:

\[ |S - S_n| \le b_{n+1}. \]

In the plot, oscillation appears in the top curve \(S_n\), while the bottom strip shows how quickly \(|a_n|\) shrinks.

3) Worked example: alternating harmonic series

The sample preset is:

\[ \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}. \]

Its sum is:

\[ \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} = \ln(2) \approx 0.693147\ldots \]

On the graph, \(S_n\) settles near \(\ln(2)\). The blinking point marks the current term \((n,a_n)\) being added.

4) Speed intuition (very brief)

Geometric (|r|<1): error shrinks rapidly like \(|r|^n\).
p-series (p>1): converges; larger p means faster shrinking terms and faster convergence.
Harmonic (1/n): diverges slowly; \(S_n\) grows roughly like \(\ln(n)\).

Frequently Asked Questions

What is a partial sum S_n in a series?

The partial sum S_n is the running total of the first n terms of a series. It is defined by S_n = sum_{k=1}^n a_k, and the difference S_n - S_{n-1} equals a_n.

How can a partial-sum plot show whether a series converges?

If the sequence of partial sums S_n approaches a finite limit as n increases, the series converges to that limit. If S_n grows without bound or fails to settle, the series diverges.

Why do some partial sums oscillate up and down?

Alternating series often produce partial sums that zig-zag because successive terms change sign. When the term magnitudes decrease toward 0, the oscillations typically shrink toward a limiting sum.

Is a_n going to 0 enough to guarantee convergence?

No. A necessary condition is a_n -> 0, but it is not sufficient; for example, the harmonic series a_n = 1/n has terms that go to 0 while its partial sums still diverge.

When does a geometric series converge and what does the visualizer show?

A geometric series with ratio r converges when |r| < 1, and its partial sums approach a finite limit. The plot shows S_n quickly stabilizing when |r| is small and failing to stabilize when |r| is at least 1.

Rate this calculator

Frequently Asked Questions

Related calculators