Examples of conditional convergence in the following topics:
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- 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
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- 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.
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- 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
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- Normality of the individual data values is not required if these conditions are met.
- Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.
- If $X_n$ converges in distribution to a random element $X$, and $Y$ converges in probability to a constant $c$, then:
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- Divergence and convergence: In the human cerebellum, information from 200 million mossy fiber inputs is expanded to 40 billion granule cells.
- This neural divergence is followed by parallel fiber outputs that converge onto 15 million Purkinje cells.
- One of the most extensively studied cerebellar learning tasks is the eyeblink conditioning paradigm.
- A blink response is elicited when a neutral conditioned stimulus, such as a tone or a light, is repeatedly paired with an unconditioned stimulus, such as an air puff.
- After
many conditioned-unconditioned stimuli (CS-US) pairings, an association is
formed whereby a learned blink, or conditioned response, occurs and precedes US onset.
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- The central limit theorem states that, given certain conditions, the mean of a sufficiently large number of independent random variables, each with a well-defined mean and well-defined variance, will be (approximately) normally distributed.
- In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions, given that they comply with certain conditions.
- By the law of large numbers, the sample averages converge in probability and almost surely to the expected value $\mu$ as $n \rightarrow \infty$.
- The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number $\mu$ during this convergence.
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- 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.
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- 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.
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- Convergent evolution occurs in different species that have evolved similar traits independently of each other.
- Convergent evolution describes the independent evolution of similar features in species of different lineages.
- They have "converged" on this useful trait.
- Convergent evolution is similar to, but distinguishable from, the phenomenon of parallel evolution.
- The opposite of convergent evolution is divergent evolution, whereby related species evolve different traits.
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- 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.