Infinite Sum Calculator - Geometric Series Sum with Steps

Two boxes, plus one optional: a = 8, r = 0.5. Click Calculate. The infinite sum calculator form speaks pure geometric series - say what each term multiplies by, and everything falls into place. No series notation needed - the infinite sum calculator reads plain numbers.

• First term (a) - the opening number of the series, 8 in 8 + 4 + 2 + 1 + \cdots. The infinite sum calculator starts counting from here.
• Common ratio (r) - the fixed multiplier between terms, 0.5 here. This is the number the infinite sum calculator tests for convergence. Type it as a decimal: 50% is 0.5.
• Terms for the partial sum (n) - optional. Fill in 6 and the infinite sum calculator adds a running-total row. Leave it empty and the infinite sum calculator skips the row.

Three rows come back from the infinite sum calculator:

• Check: converges only when |r| < 1. True means an infinite sum exists - the infinite sum calculator proceeds to give it. Feed r = 2 and this row reads false: the series diverges. That is the infinite sum calculator telling you no sum exists.
• Infinite sum S = a/(1 - r). The headline number: the total of every term. For 8 and 0.5 the infinite sum calculator returns 16.
• Partial sum of the first n terms. How close n terms get you: 8, 0.5, n = 6 return 15.75, a quarter shy of 16.

This page is more than a static infinite sum calculator - it is also an AI tutor you can ask questions:

• Ask in plain words. "My series starts at 4 and each term is one third of the last - what is the total?" The AI fills the boxes and runs the infinite sum calculator for you.
• Take a photo of your homework. The AI reads the series and fills the infinite sum calculator form.
• Ask about the ideas, not just the number. "Why does |r| < 1 decide everything?" The AI explains step by step - every number still comes from the calculator on this page, so nothing is guessed.
• Work backwards, then keep asking. Give a total and a ratio, and ask for the first term. Follow-ups are welcome - the infinite sum calculator page keeps teaching.

Can adding forever ever land on a finite number? With the right multiplier, yes. The picture behind this infinite sum calculator shows how.

Here is the series the infinite sum calculator opens with: a clean-up crew pulls 8 tons of silt in week one. Each week after, it hauls half as much as the week before:

• week 1 hauls 8 tons - running total 8
• week 2 hauls 4 - running total 12
• weeks 3 through 6 haul 2, 1, 0.5, 0.25 - totals 14, 15, 15.5, 15.75
• the totals climb ever more slowly toward 16 - the sum the infinite sum calculator returns

The stairs in the picture are those running totals - the partial sums. An infinite series is the whole staircase, $a_1 + a_2 + a_3 + \cdots$. Its value is the limit of the partial sums, if that limit exists. If the stairs never settle, the series diverges - no infinite sum exists. That limit is what the infinite sum calculator reports.

Every load here is the previous load times 0.5, so the series is geometric: $a + ar + ar^2 + ar^3 + \cdots$. That one ratio decides everything, which is why the infinite sum calculator asks for it directly. Give it 8 and 0.5, and the infinite sum calculator answers 16 - the dashed height.

Read the stairs first: up 8, up 4, up 2 - every rise is one term the infinite sum calculator adds. Now the symbols this series math calculator runs on:

$S_n = \dfrac{a(1-r^n)}{1-r}$

$S = \dfrac{a}{1-r} \quad (|r| < 1)$

The first formula adds the first n terms - the height reached so far. The second is its destination: with $|r| < 1$, the piece $ar^n$ shrinks to zero and the stairs flatten at $a/(1-r)$. Rearranged, it runs backwards: $a = S(1-r)$ recovers the first term from a known total. Both live in the infinite sum calculator above as row two and row three.

The variable names match the form boxes, so the infinite sum calculator and this page share one alphabet.

And the verdict table behind the infinite sum calculator's Check row:

Three rows in the form, three formulas here. The Check row of the infinite sum calculator reads the verdict table. The S row reads the second formula, and the partial-sum row reads the first. This series solver never guesses which case you are in - you see the verdict before you trust the number. Every number this infinite sum calculator prints traces to one of these two formulas.

The crew from the picture, in infinite sum calculator numbers. Give the infinite sum calculator its home numbers: a = 8, r = 0.5, and n = 6.

1. Check the ratio: $|0.5| < 1$ - true. The infinite sum calculator's Check row confirms the series converges.
2. Infinite sum: $S = \dfrac{8}{1-0.5} = 16$. Sixteen tons of silt in total, ever.
3. Partial sum: $S_6 = \dfrac{8(1-0.5^6)}{1-0.5} = 15.75$. Six weeks of hauling land within a quarter ton of the final 16.

The gap $16 - 15.75 = 0.25$ is the invisible tail, $0.125 + 0.0625 + \cdots$, still to come after week six. The optional n row of the infinite sum calculator makes that gap visible. Type 8, 0.5, and 6 into the infinite sum calculator above - true, 16, and 15.75 come back in one click.

Series do not always arrive as $8 + 4 + 2 + \cdots$. A calculus favorite hides the ratio inside exponents, and the infinite sum calculator still only needs $a$ and $r$:

$4 + \frac{4}{3} + \frac{4}{9} + \frac{4}{27} + \cdots = \sum_{n=1}^{\infty} 4\left(\frac{1}{3}\right)^{n-1}$

1. Match the pattern $a \cdot r^{n-1}$: the first term is $a = 4$, and each term is a third of the last, so $r = \frac{1}{3}$.
2. Check: $|\tfrac{1}{3}| < 1$, so the series converges and the infinite sum calculator gives a sum.
3. Infinite sum: $S = \dfrac{4}{1-\frac{1}{3}} = 4 \div \tfrac{2}{3} = 6$.

In the form, enter a = 4 and r = 0.333333; the infinite sum calculator returns 6 to four decimals.

Now run it backwards. A series with ratio $r = \tfrac{1}{3}$ has total sum 6 - what was the first term? Rearrange the formula: $a = S(1-r) = 6 \cdot \tfrac{2}{3} = 4$. The infinite sum calculator runs forwards; one line of algebra turns it around. Run both directions through the infinite sum calculator above when you check.

Now the trap the infinite sum calculator's Check row guards against. The series $5 + 10 + 20 + 40 + \cdots$ has $a = 5$ and $r = 2$.

1. Check: $|2| < 1$ is false - the series diverges. The infinite sum calculator answers false, and that false is the whole answer.
2. Watch the partial sums run: 5, 15, 35, 75, 155, \cdots - the infinite sum calculator has no total to report. The terms themselves grow, so no staircase is flattening.
3. Ignore step 1 and the formula "returns" $5/(1-2) = -5$: negative, from all-positive terms. The sum formula is illegal whenever $|r| \geq 1$.

Blind plugging is the classic exam slip the infinite sum calculator blocks. When the Check row reads false, do not hunt for a sum - there is none. The infinite sum calculator has done its job: it said no.

Three slips the infinite sum calculator catches every day:

1. Applying a/(1 - r) when |r| \geq 1. The sum formula holds only after the Check row of the infinite sum calculator passes. The $5 + 10 + 20 + \cdots$ example above "gives" -5: a minus sign from all-positive terms.
2. Grabbing the wrong first term. The pattern reads $a \cdot r^{n-1}$, so the exponent must match the index. In $\sum_{n=0}^{\infty} 3\left(\tfrac{1}{2}\right)^{n}$, the first term is 3, not 1.5 - the series starts at $n = 0$, where the ratio's power is zero. Match them and the infinite sum calculator gets the right first term every time.
3. Typing r as a percent. The ratio box of the infinite sum calculator takes a decimal: half is 0.5, a third is 0.333333. Entered as 50, a halving series reads as a wildly divergent one - and the Check row correctly says false.

All three slip past pencil work easily. The Check row of the infinite sum calculator above guards every one of them.