Mathwords logoMathwords

Series Rules

Series Rules

Algebra rules for convergent series are given below.

 

Three algebraic rules for convergent series: scalar multiple, sum, and difference rules using variables a_n and b_n.

 

 

See also

Sequence, arithmetic series, geometric series

Key Formula

∑n=1∞(c an)=c∑n=1∞an∑n=1∞(an±bn)=∑n=1∞an±∑n=1∞bn\sum_{n=1}^{\infty}(c\,a_n) = c\sum_{n=1}^{\infty}a_n \qquad\qquad \sum_{n=1}^{\infty}(a_n \pm b_n) = \sum_{n=1}^{\infty}a_n \pm \sum_{n=1}^{\infty}b_n
Where:
  • ana_n = The nth term of a convergent series
  • bnb_n = The nth term of another convergent series
  • cc = A constant (any real number)
  • nn = The index of summation

Worked Example

Problem: Given that the convergent series ∑(n=1 to ∞) aₙ = 5 and ∑(n=1 to ∞) bₙ = 3, find ∑(n=1 to ∞)(4aₙ − 2bₙ).
Step 1: Apply the sum/difference rule to split the series into two separate series.
∑n=1∞(4an−2bn)=∑n=1∞4an−∑n=1∞2bn\sum_{n=1}^{\infty}(4a_n - 2b_n) = \sum_{n=1}^{\infty}4a_n - \sum_{n=1}^{\infty}2b_n
Step 2: Apply the constant multiple rule to pull each constant factor outside its series.
=4∑n=1∞an−2∑n=1∞bn= 4\sum_{n=1}^{\infty}a_n - 2\sum_{n=1}^{\infty}b_n
Step 3: Substitute the known sums.
=4(5)−2(3)=20−6=14= 4(5) - 2(3) = 20 - 6 = 14
Answer: The series ∑(n=1 to ∞)(4aₙ − 2bₙ) = 14.

Another Example

This example applies the same rules to concrete geometric series rather than abstract sums, showing how the rules connect to actual evaluation.

Problem: Evaluate ∑(n=1 to ∞) [3 · (1/2)ⁿ + 5 · (1/4)ⁿ] by applying series rules to known geometric series.
Step 1: Split into two series using the sum rule.
∑n=1∞[3(12)n+5(14)n]=∑n=1∞3(12)n+∑n=1∞5(14)n\sum_{n=1}^{\infty}\left[3\left(\frac{1}{2}\right)^n + 5\left(\frac{1}{4}\right)^n\right] = \sum_{n=1}^{\infty}3\left(\frac{1}{2}\right)^n + \sum_{n=1}^{\infty}5\left(\frac{1}{4}\right)^n
Step 2: Factor out the constants from each series.
=3∑n=1∞(12)n+5∑n=1∞(14)n= 3\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^n + 5\sum_{n=1}^{\infty}\left(\frac{1}{4}\right)^n
Step 3: Each is a geometric series with first term r and ratio r, so the sum formula gives a/(1−r). For r = 1/2 the sum is (1/2)/(1−1/2) = 1. For r = 1/4 the sum is (1/4)/(1−1/4) = 1/3.
∑n=1∞(12)n=1,∑n=1∞(14)n=13\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^n = 1, \qquad \sum_{n=1}^{\infty}\left(\frac{1}{4}\right)^n = \frac{1}{3}
Step 4: Substitute back and compute.
=3(1)+5 ⁣(13)=3+53=143= 3(1) + 5\!\left(\frac{1}{3}\right) = 3 + \frac{5}{3} = \frac{14}{3}
Answer: The series equals 14/3.

Frequently Asked Questions

Do series rules work for divergent series?
No. The constant-multiple and sum/difference rules require both series to be convergent. If either series diverges, you cannot apply these rules, and the combined series may also diverge. For example, the harmonic series ∑1/n diverges, so you cannot pull out a constant and claim 2∑1/n converges.
Can you multiply two convergent series term by term?
Not simply. The product rule for series is more involved: ∑aₙ · ∑bₙ does not equal ∑(aₙbₙ). To multiply two series you need the Cauchy product, which combines terms differently. The basic series rules only cover scalar multiplication and term-by-term addition or subtraction.
Does changing the starting index of a series affect its convergence?
No. Adding or removing a finite number of terms at the beginning changes the sum's value but does not change whether the series converges or diverges. This is sometimes called the index-shift rule and is a useful companion to the main algebraic rules.

Series Rules (for series) vs. Limit Laws (for sequences)

Series Rules (for series)Limit Laws (for sequences)
What they apply toInfinite sums ∑aₙLimits of sequences lim aₙ
Constant multiple∑(caₙ) = c·∑aₙlim(caₙ) = c·lim aₙ
Sum/Difference∑(aₙ ± bₙ) = ∑aₙ ± ∑bₙlim(aₙ ± bₙ) = lim aₙ ± lim bₙ
ProductNo simple term-by-term product rule; requires Cauchy productlim(aₙ · bₙ) = lim aₙ · lim bₙ
Convergence requirementBoth series must convergeBoth sequence limits must exist (be finite)

Why It Matters

Series rules appear throughout calculus, especially when you work with power series, Taylor series, and Fourier series. They let you break a complicated series into simpler, recognizable parts you already know how to evaluate. In courses like Calculus II and beyond, these properties are used constantly to test convergence and compute exact sums.

Common Mistakes

Mistake: Applying the sum rule when one or both series diverge.
Correction: Both individual series must converge before you can add or subtract them. If ∑aₙ diverges, writing ∑(aₙ + bₙ) = ∑aₙ + ∑bₙ is invalid even if ∑bₙ converges.
Mistake: Assuming ∑(aₙ · bₙ) = (∑aₙ)(∑bₙ).
Correction: There is no term-by-term product rule analogous to the sum rule. Multiplying two series requires the Cauchy product, which sums convolution-style terms cₙ = Σ(k=0 to n) aₖbₙ₋ₖ.

Related Terms