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Series

Series

The sum of the terms of a sequence. For example, the series for the sequence 1, 3, 5, 7, 9, . . . , 131, 133 is the sum 1 + 3 + 5 + 7 + 9 + . . . + 131 + 133.

 

 

See also

Arithmetic series, geometric series, convergent series, divergent series, convergence tests, power series, positive series, series rules

Key Formula

∑k=1nak=a1+a2+a3+⋯+an\sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \cdots + a_n
Where:
  • aka_k = The k-th term of the sequence
  • nn = The number of terms being summed (for a finite series)
  • kk = The index of summation, running from 1 to n

Worked Example

Problem: Find the value of the series 2 + 4 + 6 + 8 + 10.
Step 1: Identify the sequence whose terms are being summed. The sequence is 2, 4, 6, 8, 10 — the first five positive even numbers.
ak=2kfor k=1,2,3,4,5a_k = 2k \quad \text{for } k = 1, 2, 3, 4, 5
Step 2: Write the series using summation notation.
∑k=152k=2+4+6+8+10\sum_{k=1}^{5} 2k = 2 + 4 + 6 + 8 + 10
Step 3: Add the terms together to find the sum.
2+4+6+8+10=302 + 4 + 6 + 8 + 10 = 30
Answer: The value of the series is 30.

Another Example

Problem: Write the first four partial sums of the series 1 + 1/2 + 1/4 + 1/8 + ⋯ and observe a pattern.
Step 1: The first partial sum uses only the first term.
S1=1S_1 = 1
Step 2: The second partial sum adds the first two terms.
S2=1+12=32S_2 = 1 + \tfrac{1}{2} = \tfrac{3}{2}
Step 3: The third partial sum adds the first three terms.
S3=1+12+14=74S_3 = 1 + \tfrac{1}{2} + \tfrac{1}{4} = \tfrac{7}{4}
Step 4: The fourth partial sum adds the first four terms.
S4=1+12+14+18=158S_4 = 1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} = \tfrac{15}{8}
Step 5: The partial sums are getting closer and closer to 2. This is an infinite geometric series with first term 1 and common ratio 1/2, and it converges to 2.
∑k=0∞(12)k=11−12=2\sum_{k=0}^{\infty} \left(\tfrac{1}{2}\right)^k = \frac{1}{1 - \frac{1}{2}} = 2
Answer: The partial sums approach 2, so the infinite series converges to 2.

Frequently Asked Questions

What is the difference between a series and a sequence?
A sequence is an ordered list of numbers (e.g., 1, 2, 3, 4, 5), while a series is the sum of those numbers (e.g., 1 + 2 + 3 + 4 + 5 = 15). Think of a sequence as the individual terms and a series as what happens when you add them up.
Can a series have infinitely many terms?
Yes. An infinite series adds up infinitely many terms. If the partial sums approach a finite number, the series is called convergent and that number is its sum. If the partial sums grow without bound or fail to settle on a single value, the series is called divergent.

Series vs. Sequence

A sequence is an ordered list of numbers: 3, 6, 9, 12, …. A series is the sum formed by adding those numbers: 3 + 6 + 9 + 12 + …. Every series is built from an underlying sequence, but a sequence by itself does not involve addition. You can think of a sequence as the ingredients and a series as the recipe that combines them.

Why It Matters

Series appear throughout science, engineering, and finance whenever you need to accumulate quantities — from calculating loan payments to approximating functions with Taylor series. Many functions that cannot be computed exactly, like exe^x or sin⁡x\sin x, are defined and evaluated using infinite series. Understanding series also builds the foundation for integral calculus, where continuous sums replace discrete ones.

Common Mistakes

Mistake: Using "series" and "sequence" interchangeably.
Correction: A sequence is a list of terms; a series is the sum of those terms. Writing '1, 2, 3, 4' is a sequence, while '1 + 2 + 3 + 4' is a series.
Mistake: Assuming that because the terms of a series approach zero, the series must converge.
Correction: Terms approaching zero is necessary but not sufficient for convergence. The classic counterexample is the harmonic series 1 + 1/2 + 1/3 + 1/4 + ⋯, whose terms go to zero yet the series diverges to infinity.

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