Created by Krishna Nelaturu Show
Last updated: Jul 21, 2022 Our sum of series calculator can calculate the sum of arithmetic series and geometric series. You can also use it to judge whether your series converges or diverges. If you want to learn how to find the sum of a series, you've come to the right place! Here, we shall discuss the arithmetic and geometric series and how to calculate the partial sum of these infinite series. If you want to switch to a different topic, our independent events calculator will teach you how to deal with the probabilities of three events. Arithmetic seriesConsider the sequence Arithmetic series is the sum of an arithmetic sequence. The sum of the first ten even numbers is the arithmetic series: While adding individual terms is viable for small-sized sequences, let's formulate an equation to calculate the sum of arithmetic sequences with many terms. Calculating sum of arithmetic seriesWe can find the sum of an arithmetic series if the series has a finite number of terms. The sum of the series formula for an arithmetic series is given by: Sn= n2[2a1+(n−1)d]S_n = \frac{n}{2}[2a_1 + (n-1)d] where:
Consider the sequence of the first 100 natural numbers: S100=1002[2×1+(100−1)1] =50[2+99]S100=5050\begin{align*} S_{100} &= \frac{100}{2}[2\times1 + (100-1)1]\\ &= 50[2 + 99]\\ S_{100} &= 5050 \end{align*} Geometric seriesConsider the sequence Geometric series is the sum of a geometric sequence. Before we calculate the sum of a geometric series, we must check whether the series converges or diverges. Whether an infinite geometric series converges or not depends on the common ratio
One of the examples of geometric series in physics is half life of radioactive decay which has Calculating sum of infinite geometric seriesWhen the common ratio S=a1− rS = \frac{a}{1-r} where:
Consider the infinite geometric series: S =a1−r=11−110=1910S=109=1.1‾\begin{align*} S &= \frac{a}{1-r} = \frac{1}{1-\frac{1}{10}}\\ &=\frac{1}{\frac{9}{10}}\\ S &= \frac{10}{9} = 1.\overline{1} \end{align*} The line over the decimal digit indicates a non-terminating repeating decimal. Calculating partial sum of geometric seriesThe 2k geometric series ( Sn=a(1−rn)1−rS_n = \frac{a(1-r^n)}{1-r} where:
The sum of the first 10 terms of the 2k geometric series would be: S10=2(1−210 )1−2=−2(1−1024)S10 =2046\begin{align*} S_{10} &= \frac{2(1-2^{10})}{1-2}\\ &= -2(1-1024)\\ S_{10} &=2046 \end{align*} Now you know how to find the sum of any series. How to use this sum of series calculatorThis sum of a series calculator makes it easy to find the sum of an arithmetic series or a geometric series. To calculate the sum of an arithmetic sequence:
To calculate the sum of a geometric series:
How do you find the sum of a finite geometric series?To find the sum of a finite geometric series, use the formula, Sn=a1(1−rn)1−r,r≠1 , where n is the number of terms, a1 is the first term and r is the common ratio .
What is the sum of the finite arithmetic series calculator?Sn = (n/2)×[2a + (n-1)×d] is the formula to find the sum of n terms of an arithmetic progression, where: n is the number of terms; a is the first term; and. d is the common difference, or the difference between successive terms.
How do you find the sum of an infinite geometric series on a calculator?The sum of a series Sn is calculated using the formula Sn=a(1−rn)1−r S n = a ( 1 - r n ) 1 - r . For the sum of an infinite geometric series S∞ , as n approaches ∞ , 1−rn 1 - r n approaches 1 .
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