conditional convergence

Examples of conditional convergence in the following topics:

• Tips for Testing Series

  • Convergence tests are methods of testing for the convergence or divergence of an infinite series.
  • Convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence, or divergence of an infinite series.
  • When testing the convergence of a series, you should remember that there is no single convergence test which works for all series.
  • Here is a summary for the convergence test that we have learned:
  • Formulate three techniques that will help when testing the convergence of a series

• Convergence of Series with Positive Terms

  • The series $\sum_{n \ge 1} \frac{1}{n^2}$ is convergent because of the inequality:
  • converge?
  • It is possible to "visualize" its convergence on the real number line?
  • For these specific examples, there are easy ways to check the convergence.
  • However, it could be the case that there are no easy ways to check the convergence.

• Absolute Convergence and Ratio and Root Tests

  • An infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute value of the summand is finite.
  • (A convergent series that is not absolutely convergent is called conditionally convergent.)
  • The root test is a criterion for the convergence (a convergence test) of an infinite series.
  • otherwise the test is inconclusive (the series may diverge, converge absolutely, or converge conditionally).
  • State the conditions when an infinite series of numbers converge absolutely

• Comparison Tests

  • The limit comparison test is a method of testing for the convergence of an infinite series, while the direct comparison test is a way of deducing the convergence or divergence of an infinite series or an improper integral by comparison with other series or integral whose convergence properties are already known.
  • Example: We want to determine if the series $\Sigma \frac{n+1}{2n^2}$ converges or diverges.
  • In both cases, the test works by comparing the given series or integral to one whose convergence properties are known.
  • If the infinite series $\sum b_n$ converges and $0 \le a_n \le b_n$ for all sufficiently large $n$ (that is, for all $n>N$ for some fixed value $N$), then the infinite series $\sum a_n$ also converges.
  • The series $\Sigma \frac{1}{n^3 + 2n}$ converges because $\frac{1}{n^3 + 2n} < \frac{1}{n^3}$ for $n > 0$ and $\Sigma \frac{1}{n^3}$ converges.

• Summing an Infinite Series

  • Infinite sequences and series can either converge or diverge.
  • A series is said to converge when the sequence of partial sums has a finite limit.
  • By definition the series $\sum_{n=0}^\infty a_n$ converges to a limit $L$ if and only if the associated sequence of partial sums  converges to $L$.
  • An easy way that an infinite series can converge is if all the $a_{n}$ are zero for sufficiently large $n$s.
  • This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy.

• Alternating Series

  • Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.
  • The theorem known as the "Leibniz Test," or the alternating series test, tells us that an alternating series will converge if the terms $a_n$ converge to $0$ monotonically.
  • Similarly, it can be shown that, since $a_m$ converges to $0$, $S_m - S_n$ converges to $0$ for $m, n \rightarrow \infty$.
  • Therefore, our partial sum $S_m$ converges.
  • $a_n = \frac1n$ converges to 0 monotonically.

• The Integral Test and Estimates of Sums

  • The integral test is a method of testing infinite series of nonnegative terms for convergence by comparing them to an improper integral.
  • The integral test for convergence is a method used to test infinite series of non-negative terms for convergence.
  • The infinite series $\sum_{n=N}^\infty f(n)$ converges to a real number if and only if the improper integral $\int_N^\infty f(x)\,dx$ is finite.
  • On the other hand, the series $\sum_{n=1}^\infty \frac1{n^{1+\varepsilon}}$ converges for every $\varepsilon > 0$ because, by the power rule:
  • In this way, it is possible to investigate the borderline between divergence and convergence of infinite series.

• Power Series

  • A power series will converge for some values of the variable $x$ and may diverge for others.
  • All power series $f(x)$ in powers of $(x-c)$ will converge at $x=c$.
  • If $c$ is not the only convergent point, then there is always a number $r$ with 0 < r ≤ ∞ such that the series converges whenever $\left| x-c \right| r$.
  • The number $r$ is called the radius of convergence of the power series.

• Series

  • By definition, the series $\sum_{n=0}^{\infty} a_n$ converges to a limit $L$ if and only if the associated sequence of partial sums $\{S_k\}$ converges to $L$.
  • State the requirements for a series to converge to a limit

• Sequences

  • The plot of a convergent sequence ($a_n$) is shown in blue.
  • Visually, we can see that the sequence is converging to the limit of $0$ as $n$ increases.