And then when n isĮqual to 5, a sub n is equal to positive 1/5, which N is equal to negative 1/4, which is right about there. Sub n is equal to 1/3, which is right about there. When n is equal to 1, a sub n is equal to 1. Vertical and horizontal axes at the same scale, just so that Horizontal axis where I'm going to plot our n's. And it just keeps going onĪnd on and on like this. infinity + 1 = infinity, infinity^2 = infinity / 8, basically infinity can't be dealt with in algebra, (although L'Hopitol's rule lets us manage it), and shouldn't be thought of as a "number" in the common sense of the word. And remember, infinity is MUCH larger than 36, and this function will continue to leave potential values in the dust, never looking back, indefinitely.Īlso, just in case it isn't clear, infinity doesn't follow many of the properties that numbers follow. ![]() We might claim that it goes to 34359738368, a very big number equal to 2^35, but we 2^x, being fickle to the point of cruelty, will leave this value (already large) for a notably larger one as x goes to 36. What number does 2^x go to? (It diverges) If we were to investigate sin(x)/x, it would converge at 0, because the dividing by x heads to 0, and the +/- 1 can't stop it's approach.Ī similar resistance to staying mostly still can be found in equations that diverge as their inputs approach infinity. They don't head to infinity, and they don't converge. But, we know that they will always fluctuate. We know they will never output anything greater than 1, or less than -1, we are even able to compute them for any real number. One of the main things a function has to do to approach a number is to start to stabilize. Not all functions approach a number as their input approaches infinity. So we have an indeterminate form when we have a base approaching 1 and exponent approaching infinity, but not when we have a base that EQUALS 1 and exponent approaching infinity. We no longer have an infinitesimal increment away from 1 that can be overpowered by the increase of the exponent. The expression 1^b is always 1, no matter how large or small the exponent. HOWEVER, if a is not some function that approaches 1, but is actually the number 1, then we no longer have an indeterminate form. So that is why we say 1^∞ is an indeterminate form. If we find that a approaches 1 and b approaches infinity, we have an indeterminate form, because we can't tell without further analysis whether the forces attracting a toward 1 (making the expression approach 1) are overpowered by the forces moving b toward infinity (making the expression approach infinity or zero, depending on whether a is slightly greater than or less than 1). ![]() In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be computed by successive applications of the recurrence relation.Suppose we want to know the limit of a^b as x goes to infinity, where a and b are both functions of x. To define a sequence by recursion, one needs a rule, called recurrence relation to construct each element in terms of the ones before it. This is in contrast to the definition of sequences of elements as functions of their positions. Sequences whose elements are related to the previous elements in a straightforward way are often defined using recursion. In mathematical analysis, a sequence is often denoted by letters in the form of a n a_, but it is not the same as the sequence denoted by the expression.ĭefining a sequence by recursion The first element has index 0 or 1, depending on the context or a specific convention. The position of an element in a sequence is its rank or index it is the natural number for which the element is the image. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6. Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set.įor example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. The number of elements (possibly infinite) is called the length of the sequence. Like a set, it contains members (also called elements, or terms). ![]() In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. For other uses, see Sequence (disambiguation). For the sequentional logic function, see Sequention. For the manual transmission, see Sequential manual transmission.
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