Integration and series · Definition

Numerical series

real analysisseries

Before you start you need to know

It gives meaning to adding infinitely many numbers: instead of adding them all at once, we study the limit of the finite sums as they accumulate.

Definition

A series is the sequence of partial sums \(s_N\) of a sequence of terms \(a_n\); it converges if \(s_N\) has a finite limit, and that limit is its sum.

\[s_N=\sum_{n=1}^{N}a_n,\qquad \sum_{n=1}^{\infty}a_n=\lim_{N\to\infty}s_N\]

Adding infinitely many terms is not an arithmetic operation but a limit. We form the partial sums \(s_1=a_1,\ s_2=a_1+a_2,\ \dots\) and ask whether that sequence converges. Everything you know about sequences carries over: the series converges exactly when \((s_N)\) converges, and its sum is the limit.

Two families serve as guiding examples. The geometric series \(\sum_{n=0}^{\infty}r^n\) converges to \(\frac1{1-r}\) if \(|r|<1\) and diverges if \(|r|\ge1\). Telescoping series, such as \(\sum\frac1{n(n+1)}\), are computed by rewriting each term as a difference so that almost everything cancels and \(s_N=1-\frac1{N+1}\to1\).

A necessary condition: if \(\sum a_n\) converges, then \(a_n=s_n-s_{n-1}\to0\). It is only necessary. The harmonic series \(\sum\frac1n\) has terms tending to zero and still diverges, because its partial sums grow without bound. Concluding convergence just because \(a_n\to0\) is the most common mistake.

Since convergence is that of a sequence, there is also a Cauchy version: \(\sum a_n\) converges if and only if the blocks \(a_{m+1}+\dots+a_n\) become small. The next tests (comparison, absolute convergence) are practical ways to decide this without computing the sum.

Think about it

If \(a_n\to0\), can \(\sum a_n\) diverge? Give an example.

Yes: in \(\sum\frac1n\) the term tends to zero, but the partial sums grow without bound.

For which values of \(r\) does \(\sum_{n=0}^{\infty}r^n\) converge?

For \(|r|<1\), with sum \(\frac1{1-r}\); otherwise the term does not tend to zero.

Exercises 10

  1. Compute \(\displaystyle\sum_{n=0}^{\infty}\left(\tfrac12\right)^n\).
  2. Compute \(\displaystyle\sum_{n=1}^{\infty}\frac{1}{n(n+1)}\).
  3. For the series \(\sum_{n=1}^{\infty}\frac1{n^2}\), compute the partial sum \(s_3\).
  4. What happens with \(\displaystyle\sum_{n=0}^{\infty}(-1)^n\)?
  5. Which of these series diverges because its general term does not tend to zero?
  6. Compute \(\displaystyle\sum_{n=1}^{\infty}\frac{3}{4^n}\).
  7. Compute \(\displaystyle\sum_{n=2}^{\infty}\frac{1}{n^2-1}\).
  8. Compute \(\displaystyle\sum_{n=0}^{\infty}\left(-\tfrac23\right)^n\).
  9. The partial sums of a series are \(s_N=\dfrac{2N}{N+1}\). What is the term \(a_3\)?
  10. Which statement about \(\sum a_n\) is true?

References

  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §2.7
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 3, series
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §3.7