Examples of series in the following topics:
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- A series is the sum of the terms of a sequence.
- Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
- Infinite sequences and series can either converge or diverge.
- Working out the properties of the series that converge even if infinitely many terms are non-zero is, therefore, the essence of the study of series.
- In the following atoms, we will study how to tell whether a series converges or not and how to compute the sum of a series when such a value exists.
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- When testing the convergence of a series, you should remember that there is no single convergence test which works for all series.
- It is up to you to guess and pick the right test for a given series.
- But if the integral diverges, then the series does so as well.
- Direct comparison test: If the series $\sum_{n=1}^\infty b_n$ is an absolutely convergent series and $\left |a_n \right | \le \left | b_n \right|$ for sufficiently large $n$, then the series $\sum_{n=1}^\infty a_n$ converges absolutely.
- The integral test applied to the harmonic series.
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- If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
- A function may not be equal to its Taylor series, even if its Taylor series converges at every point.
- In the case that $a=0$, the series is also called a Maclaurin series.
- The Maclaurin series for $(1 − x)^{−1}$ for $\left| x \right| < 1$ is the geometric series: $1+x+x^2+x^3+\cdots\!
- Identify a Maclaurin series as a special case of a Taylor series
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- A power series (in one variable) is an infinite series of the form $f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n$, where $a_n$ is the coefficient of the $n$th term and $x$ varies around $c$.
- A power series (in one variable) is an infinite series of the form:
- This series usually arises as the Taylor series of some known function.
- can be written as a power series around the center $c=1$ as:
- In such cases, the power series takes the simpler form
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- A series is the sum of the terms of a sequence.
- A series is, informally speaking, the sum of the terms of a sequence.
- Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
- An example is the famous series from Zeno's dichotomy and its mathematical representation:
- State the requirements for a series to converge to a limit
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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.
- If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$ with $0 < c < \infty$, then either both series converge or both series diverge.
- For this we compare it with the series $\Sigma \frac{1}{n}$, which diverges.
- In this atom, we will check the series case only.
- If the infinite series $\sum b_n$ diverges and $a_n \ge b_n \ge 0$ for all sufficiently large $n$, then the infinite series $\sum a_n$ also diverges.
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- The power series method is used to seek a power series solution to certain differential equations.
- The power series method is used to seek a power series solution to certain differential equations.
- The power series method calls for the construction of a power series solution:
- The series solution is:
- Identify the steps and describe the application of the power series method
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- A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of the function's derivatives.
- The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable in a neighborhood of a real or complex number $a$ is the power series
- Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.
- To evaluate the integral $I = \int_{a}^{b} f(x) \, dx$, we can Taylor-expand $f(x)$ and perform integration on individual terms of the series.
- The exponential function (in blue) and the sum of the first 9 terms of its Taylor series at 0 (in red).
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- An alternating series is an infinite series of the form $\sum_{n=0}^\infty (-1)^n\,a_n$ or $\sum_{n=0}^\infty (-1)^{n-1}\,a_n$ with $a_n > 0$ for all $n$.
- 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.
- Therefore, the sum $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ converges by the alternating series test.
- Describe the properties of an alternating series and the requirements for one to converge
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- It is particularly useful in connection with power series.
- if $C = 1$ and the limit approaches strictly from above, then the series diverges;
- There are some series for which $C = 1$ and the series converges, e.g.:
- and there are others for which $C = 1$ and the series diverges, e.g.:
- State the conditions when an infinite series of numbers converge absolutely