HexMathv1.0.3

Partial sum of series calculator

Type a sum with its limits. HexMath adds the first terms and gives the exact partial sum.

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What this page covers

  • Partial sums of arithmetic and geometric series, using their sum formulas.
  • Telescoping sums, where most terms cancel.
  • Short sums of any terms, added exactly as fractions.
  • Not covered here: whether an infinite series converges (use the series convergence calculator), and sums with no closed form beyond adding the terms.

How to enter the problem

  • Type it or paste it. Write the sum with sigma notation, the starting value and the last value, for example sum from n = 1 to 10 of 1/(n(n+1)).
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

Find ∑k=120(3k−1)\displaystyle \sum_{k=1}^{20} (3k - 1).

Answer

Verified

The answer is

S20=610S_{20} = 610

Explanation

  1. Recognise the series
    The terms 2,5,8,…2, 5, 8, \dots go up by 3, so this is arithmetic.
  2. Find the first and last terms
    First 22, last 3(20)−1=593(20) - 1 = 59.
  3. Use the arithmetic sum
    S20=202(2+59)=10⋅61=610S_{20} = \frac{20}{2}(2 + 59) = 10 \cdot 61 = 610

Example 2

Problem

Find ∑n=165(12)n−1\displaystyle \sum_{n=1}^{6} 5\left(\frac{1}{2}\right)^{n-1}.

Answer

Verified

The answer is

31532\frac{315}{32}

Explanation

  1. Recognise the series
    Each term is half the one before: geometric with first term a=5a = 5 and ratio r=12r = \frac{1}{2}.
  2. Use the geometric sum
    S6=a1−r61−r=5⋅1−16412S_6 = a\frac{1 - r^6}{1 - r} = 5 \cdot \frac{1 - \frac{1}{64}}{\frac{1}{2}}
  3. Simplify
    5⋅2⋅6364=63064=315325 \cdot 2 \cdot \frac{63}{64} = \frac{630}{64} = \frac{315}{32}

Example 3

Problem

Find ∑n=18(−1)n+1n\displaystyle \sum_{n=1}^{8} \frac{(-1)^{n+1}}{n}.

Answer

Verified

The answer is

533840\frac{533}{840}

Explanation

  1. Write out the terms
    1−12+13−14+15−16+17−181 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} - \frac{1}{8}.
  2. Use a common denominator
    The least common multiple of 1 to 8 is 840: 840−420+280−210+168−140+120−105840\frac{840 - 420 + 280 - 210 + 168 - 140 + 120 - 105}{840}.
  3. Add the numerators
    533840\frac{533}{840}, about 0.63450.6345.

Common mistakes

  • Counting one term too many. Stopping at n=11n = 11 instead of n=10n = 10 in the example gives 1112\frac{11}{12} instead of 1011\frac{10}{11}.
  • Using rnr^{n} in place of rn−1r^{n-1} when the sum starts at n=1n = 1, which changes the first term.
  • In a telescoping sum, keeping the cancelled middle terms or dropping the last one that remains.
  • Losing the alternating signs in sums with (−1)n+1(-1)^{n+1}.

Checks, assumptions and limits

  • Check by hand: add the first two or three terms yourself and compare with the formula for S2S_2 or S3S_3.
  • A partial sum is the sum of a fixed number of terms. It exists even when the infinite series does not converge.
  • HexMath can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

It is the total of the first n terms of a series, often written S_n. As n grows, partial sums show whether an infinite series settles to a value.

Yes. Partial sums of fractions are kept exact. You can ask for a decimal value as a follow-up.

For arithmetic, geometric and telescoping series, yes. Ask for S_n in terms of n and the steps show where it comes from.

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