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Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.$ \displaystyle \sum_{n = 1}^{\infty} \frac {5}{\pi^n} $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 2
Series
Sequences
Ethan G.
November 16, 2020
what test or method were you using?
Missouri State University
Idaho State University
Boston College
Lectures
01:59
In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.
02:28
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.
01:27
Determine whether the geom…
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Determine whether the seri…
02:17
interment weather this geometric Siri's converges. And if it does converge, let's go ahead and find the song. So this Siri's We could go ahead and rewrite this to see if it's geometric. Well not. Not only we know it's your metric they told us that, but if we go ahead and rewrite this as one or a pie to the O. R. Teo, excuse me to the end. We now have it in the form a r n. And so here we can use the formula. So the formula is you take the first term in the sun and then you divide it by one minus are the common ratio. Hear our case, our equals one over pi. Then I think about this formula here is it doesn't matter what you're starting value for end is and it doesn't matter what the exponents is, for example, could be and plus one and minus three. This formula will always work for geometric assuming, of course, we only use this when it converges. And that's when absolute value are is less than one equivalently negative, one less than are less than one in our problem. This value satisfies that it's positive so it's bigger than negative one, but it's more or less one over three, so that's definitely less than one. So this will converge. And then if we use the formula while the first term here the first term, when is when you plug in and equals one? See you get five over pi and then we have one minus R so one minus one over pi And if we like, we could just go ahead and simple by this five over pi up top and then on the bottom, pine minus one over pi and we end up after cancelling those pies. We have five time itis, one on the bottom and that's your final answer.
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