If you have to change the amplitude, period, and position of a secant or cosecant graph, your best bet is to graph their reciprocal functions and transform them first. The reciprocal functions, sine and cosine, are easier to graph because they don’t have as many complex parts (no asymptotes, basically). If you can graph the.
If you have to change the amplitude, period, and position of a secant or cosecant graph, your best bet is to graph their reciprocal functions and transform them first. The reciprocal functions, sine and cosine, are easier to graph because they don’t have as many complex parts (no asymptotes, basically). If you can graph the reciprocals first, you can deal with the more complicated pieces of the secant/cosecant graphs last.
For example, take a look at the graph
- Graph the transformed reciprocal function y = 1/4 cos x – 1.Look at the reciprocal function for secant, which is cosine. Pretend just for a bit that you’re graphingFollow all the rules for the cosine graph in order to end up with a graph that looks like the one in the figure.
- Sketch the asymptotes of the transformed reciprocal function.Wherever the transformed graph involving cosine crosses its sinusoidal axis, you have an asymptote in the graph involving secant. You see that the cosine graph crosses the sinusoidal axis when x = pi/2 and 3pi/2.
- Find out what the graph looks like between each asymptote.Now that you’ve identified the asymptotes, you simply figure out what happens on the intervals between them. The finished graph,ends up looking like the one in the figure.
- State the domain and range of the transformed function.Because the new transformed function may have different asymptotes than the parent function for secant and it may be shifted up or down, you may be required to state the new domain and range.This example,Therefore, the domain is restricted to not include these values and is writtenwhere x is an integer. In addition, the range of this function changes because the transformed function is shorter than the parent function and has been shifted down 2. The range has two separate intervals,
You can graph a transformation of the cosecant graph by using the same steps you use when graphing the secant function, only this time you use the sine function to guide you.
The shape of the transformed cosecant graph should be very similar to the secant graph, except the asymptotes are in different places. For this reason, be sure you’re graphing with the help of the sine graph (to transform the cosecant graph) and the cosine function (to guide you for the secant graph).
For example, graph the transformed cosecant graph
- Graph the transformed reciprocal function.Look first at the functionThe rules to transforming a sine function tell you to first factor out the 2 and getIt has a horizontal shrink of 2, a horizontal shift ofto the right, and a vertical shift of up 1. The figure shows the transformed sine graph.
- Sketch the asymptotes of the reciprocal function.The sinusoidal axis that runs through the middle of the sine function is the line y = 1. Therefore, an asymptote of the cosecant graph exists everywhere the transformed sine function crosses this line. The asymptotes of the graph involving cosecant are at
- Figure out what happens to the graph between each asymptote.You can use the transformed graph of the sine function to determine where the cosecant graph is positive and negative. Because the graph of the transformed sine function is positive in betweenthe cosecant graph is positive as well and extends up when getting closer to the asymptotes. Similarly, because the graph of the transformed sine function is negative in betweenthe cosecant is also negative in this interval. The graph alternates between positive and negative in equal intervals forever in both directions.The figure shows the transformed cosecant graph.
- State the new domain and range.Just as with the transformed graph of the secant function, you may be asked to state the new domain and range for the cosecant function. The domain of the transformed cosecant function is all values of x except for the values that are asymptotes. From the graph, you can see that the domain is all values of x, wherewhere x is an integer. The range of the transformed cosecant function is also split up into two intervals:
In mathematics, a series is the sum of the terms of an infinite sequence of numbers.
Given an infinite sequence , the nth partial sum is the sum of the first n terms of the sequence, that is,
A series is convergent if the sequence of its partial sums tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number such that for every arbitrarily small positive number , there is a (sufficiently large) integer such that for all ,
If the series is convergent, the number (necessarily unique) is called the sum of the series.
Any series that is not convergent is said to be divergent.
Examples of convergent and divergent series[edit]
- The reciprocals of the positive integers produce a divergent series (harmonic series):
- Alternating the signs of the reciprocals of positive integers produces a convergent series:
- The reciprocals of prime numbers produce a divergent series (so the set of primes is 'large'):
- The reciprocals of triangular numbers produce a convergent series:
- The reciprocals of factorials produce a convergent series (see e):
- The reciprocals of square numbers produce a convergent series (the Basel problem):
- The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is 'small'):
- The reciprocals of powers of any n>1 produce a convergent series:
- Alternating the signs of reciprocals of powers of 2 also produces a convergent series:
- Alternating the signs of reciprocals of powers of any n>1 produces a convergent series:
- The reciprocals of Fibonacci numbers produce a convergent series (see ψ):
Convergence tests[edit]
There are a number of methods of determining whether a series converges or diverges.
![Reciprocal Graph Convergence Reciprocal Graph Convergence](/uploads/1/2/3/9/123900935/840230666.png)
If the blue series, , can be proven to converge, then the smaller series, must converge. By contraposition, if the red series is proven to diverge, then must also diverge.
Comparison test. The terms of the sequence are compared to those of another sequence . If,
for all n, , and converges, then so does
However, if,
for all n, , and diverges, then so does
Ratio test. Assume that for all n, is not zero. Suppose that there exists such that
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.
Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:
![Reciprocal Graph Convergence Reciprocal Graph Convergence](/uploads/1/2/3/9/123900935/841512196.png)
- where 'lim sup' denotes the limit superior (possibly ∞; if the limit exists it is the same value).
If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.
The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true. The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series.
Integral test. The series can be compared to an integral to establish convergence or divergence. Let be a positive and monotonically decreasing function. If
then the series converges. But if the integral diverges, then the series does so as well.
Limit comparison test. If , and the limit exists and is not zero, then converges if and only if converges.
Alternating series test. Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form , if is monotonically decreasing, and has a limit of 0 at infinity, then the series converges.
Cauchy condensation test. If is a positive monotone decreasing sequence, then converges if and only if converges.
Conditional and absolute convergence[edit]
Illustration of the absolute convergence of the power series of Exp[z] around 0 evaluated at z = Exp[i⁄3]. The length of the line is finite.
Illustration of the conditional convergence of the power series of log(z+1) around 0 evaluated at z = exp((π−1⁄3)i). The length of the line is infinite.
For any sequence , for all n. Therefore,
This means that if converges, then also converges (but not vice versa).
If the series converges, then the series is absolutely convergent. An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long. The power series of the exponential function is absolutely convergent everywhere.
If the series converges but the series diverges, then the series is conditionally convergent. The path formed by connecting the partial sums of a conditionally convergent series is infinitely long. The power series of the logarithm is conditionally convergent.
The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.
Uniform convergence[edit]
Let be a sequence of functions. The series is said to converge uniformly to fif the sequence of partial sums defined by
converges uniformly to f.
There is an analogue of the comparison test for infinite series of functions called the Weierstrass M-test.
Cauchy convergence criterion[edit]
The Cauchy convergence criterion states that a series
converges if and only if the sequence of partial sums is a Cauchy sequence.This means that for every there is a positive integer such that for all we have
which is equivalent to
See also[edit]
External links[edit]
- Hazewinkel, Michiel, ed. (2001) [1994], 'Series', Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN978-1-55608-010-4
- Weisstein, Eric (2005). Riemann Series Theorem. Retrieved May 16, 2005.
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Convergent_series&oldid=898324910'