Sequences are always either monotonic or not monotonic. If a sequence is monotonic, it means that it’s always increasing or always decreasing. If a sequence is sometimes increasing and sometimes decreasing and therefore doesn’t have a consistent direction, it means that the sequence is not monotonic. In other words, a non-monotonic sequence is increasing for parts of the sequence and decreasing for others.
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A normal series is given by a_n, where a_n is the sequence whose n values increase by increments of 1. On the other hand, a partial sums sequence is called s_n, and its n values increase by additive increments. This means that the first term in a partial sums sequence is the n=1 term, the second term is the n=1 term plus the n=2 term, the third term is (n=1)+(n=2)+(n=3), etc.
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Telescoping series are series in which all but the first and last terms cancel out. If you think about the way that a long telescope collapses on itself, you can better understand how the middle of a telescoping series cancels itself. To determine whether a series is telescoping, we’ll need to calculate at least the first few terms to see whether the middle terms start canceling with each other.
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We can use the p-series test for convergence to say whether or not a_n will converge. The p-series test says that a_n will converge when p>1 but that a_n will diverge when p≤1. The key is to make sure that the given series matches the format above for a p-series, and then to look at the value of p to determine convergence.
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When the terms of a series decrease toward 0, we say that the series is converging. Otherwise, the series is diverging. The nth term test is inspired by this idea, and we can use it to show that a series is diverging. Ironically, even though the nth term test is one of the convergence tests that we learn when we study sequences and series, it can only test for divergence, it can never confirm convergence.
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The ratio test for convergence lets us determine the convergence or divergence of a series a_n using a limit, L. Once we find a value for L, the ratio test tells us that the series converges absolutely if L<1, and diverges if L>1 or if L is infinite. The test is inconclusive if L=1. The ratio test is used most often when our series includes a factorial or something raised to the nth power.
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The interval of convergence of a series is the set of values for which the series is converging. Remember, even if we can find an interval of convergence for a series, it doesn’t mean that the entire series is converging, only that the series is converging in the specific interval. The radius of convergence of a series is always half of the interval of convergence. You can remember this if you think about the interval of convergence as the diameter of a circle.
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The arc length of a polar curve is simply the length of a section of a polar parametric curve between two points a and b. We use a specific formula in terms of L, the arc length, r, the equation of the polar curve, (dr/dtheta), the derivative of the polar curve, and a and b, the endpoints of the section.
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We can use the formula for the sum of a geometric series to quickly and accurately convert a repeating decimal into a ratio of integers, in other words, into a fraction with whole numbers in the numerator and denominator.
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This theorem looks elaborate, but it’s nothing more than a tool to find the remainder of a series. For example, oftentimes we’re asked to find the nth-degree Taylor polynomial that represents a function f(x). The sum of the terms after the nth term that aren’t included in the Taylor polynomial is the remainder. We can use Taylor’s inequality to find that remainder and say whether or not the nth-degree polynomial is a good approximation of the function’s actual value.
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Taylor series let us find a series representation for any function. In order to create a Taylor series representation for a function, we’ll need ‘a’, the value about which the function is defined and ‘n’, the degree to which we want to evaluate the function. Both of these are usually given in the problem. With a value for ‘a’ and ‘n’, we can build the chart below.
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The alternating series test for convergence lets us say whether an alternating series is converging or diverging. When we use the alternating series test, we need to make sure that we separate the series a_n from the (-1)^n part that makes it alternating.
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Only monotonic sequences can be bounded, because bounded sequences must be either increasing or decreasing, and monotonic sequences are sequences that are always increasing or always decreasing.
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Sometimes we’ll want to use polynomial long division to simplify a fraction, but either the numerator and/or denominator isn’t a polynomial. In this case, we may be able to replace the non-polynomial with its power series expansion, which will be a polynomial. The simplest way to do this for the non-polynomial is to find a similar, known power series expansion and then modify it to match the non-polynomial function.
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To calculate the work done when we lift a weight or mass vertically some distance, we’ll use the integration formula for work, where W is the work done, F(x) is the force equation, and [a,b] is the starting and ending height of the weight or mass. Oftentimes problems like these will have us use a rope or cable to lift an object up some vertical height. In a problem like this, we’ll need to determine the combined force required to lift the rope and the object.
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The general term of a sequence an is a term that can represent every other term in the sequence. It relates each term in the sequence to its place in the sequence. To find the general term, a_n, we need to relate the pattern in the sequence of terms to the corresponding value of n.
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Improper integrals are just like definite integrals, except that the lower and/or upper limit of integration is infinite. Remember that a definite integral is an integral that we evaluate over a certain interval. An improper integral is just a definite integral where one end of the interval is +/-infinity.
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The Mean Value Theorem for integrals tells us that, for a continuous function f(x), there’s at least one point c inside the interval [a,b] at which the value of the function will be equal to the average value of the function over that interval. This means we can equate the average value of the function over the interval to the value of the function at the single point.
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The goal is to use differentiation to get the left side of this equation to match exactly the function we’ve been given. When we differentiate, we have to remember to differentiate all three parts of the equation. We’ll try to simplify the sum on the right as much as possible, and the result will be the power series representation of our function. If we need to, we can then use the power series representation to find the radius and interval of convergence.
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Previously we learned how to create a power series representation for a function by modifying a similar, known series to match the function. When we have the product of two known power series, we can find their product by multiplying the expanded form of each series in the product.
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