Need help with a (apparently) difficult series

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In summary, the conversation discusses the use of small angle approximations in dealing with an infinite series involving trigonometric functions. The speaker mentions the need for more than just the bound on ##\sin(n)## and suggests using the Taylor series expansions for the approximations of ##\sin(1/n)## and ##\cos(1/\sqrt{n})##. However, there are concerns about the convergence rate of the approximations in an infinite series.
  • #1
Sebacide
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Homework Statement
I have a series, but i cannot find a method to study the convergence. Can anyone, please, help me with this series? I can't understand what kind of method can be used to study its convergence.
Relevant Equations
I thought about using ##0\leq\left|\sin(n)\right|\leq1##
This is the series: $$\sum_{n=1}^{+\infty}\sin(n)\sin\left(\frac{1}{n}\right)\left(\cos\left(\frac{1}{\sqrt{n}}\right)-1\right)$$
 
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  • #2
Good start. I'm sure that you will need to use the bound on ##\sin(n)## that you mention, but you need more than that. I think it must have something to do with the small angle approximations. For large ##n##, ##\sin(1/n) \approx 1/n## and ##\cos(1/\sqrt{n}) \approx 1- \frac{1}{2n}##. These approximations can be derived from the Taylor series expansions. But I have concerns about using approximations in an infinite series since I do not know how fast the terms approach their approximate value.
 

FAQ: Need help with a (apparently) difficult series

What is a series in mathematics?

A series in mathematics is the sum of the terms of a sequence. It can be either finite, with a limited number of terms, or infinite, where terms continue indefinitely. Series are often represented using sigma notation and can be convergent (approaching a specific value) or divergent (growing without bound).

How do I determine if a series is convergent or divergent?

To determine if a series is convergent or divergent, you can use various tests such as the comparison test, ratio test, root test, integral test, and alternating series test. Each test has specific criteria that help assess the behavior of the series as the number of terms increases.

What is the difference between a sequence and a series?

A sequence is an ordered list of numbers, where each number is called a term. A series, on the other hand, is the sum of the terms of a sequence. While a sequence lists the terms individually, a series adds them together to form a single value or expression.

What are common types of series?

Common types of series include arithmetic series, geometric series, harmonic series, and power series. Each type has specific characteristics and formulas for finding the sum of its terms. For example, a geometric series has a constant ratio between consecutive terms, while an arithmetic series has a constant difference.

Can a divergent series have a finite sum?

No, a divergent series does not have a finite sum. By definition, a divergent series grows without bound or oscillates without settling to a specific value. Only convergent series have finite sums, where the partial sums approach a specific limit as the number of terms increases.

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