What are infinite series used for?
It is very tough to answer such a general question, but I will give it a shot. Infinite series allow us to add up infinitely many terms, so it is suitable for representing something that keeps on going forever; for example, a geometric series can be used to find a fraction equivalent to any given repeating decimal such as:
by splitting into individual decimals,
by rewriting into a form of geometric series,
by using the formula for the sum of geometric series,
The knowledge of geometric series helped us find the fraction fairly easily. I hope that this was helpful.
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Infinite series are used in various fields of mathematics, science, and engineering for several purposes, including:

Calculating mathematical constants: Infinite series can be used to compute mathematical constants such as π (pi), e (Euler's number), and others to a desired level of precision.

Approximating functions: Infinite series can represent functions as sums of infinitely many terms, allowing for the approximation of functions within certain intervals or conditions.

Solving differential equations: Infinite series can be employed to find solutions to differential equations, particularly in cases where exact analytical solutions are difficult to obtain.

Analyzing sequences and series: Infinite series provide a framework for studying the convergence or divergence of sequences and series, which is fundamental in mathematical analysis.

Signal processing: In engineering and physics, infinite series are used in signal processing to analyze and manipulate signals, particularly in Fourier analysis where signals are represented as infinite series of sinusoidal functions.

Probability and statistics: Infinite series play a role in probability theory and statistics, particularly in the study of random processes and the calculation of probabilities.

Financial mathematics: Infinite series are utilized in finance for the valuation of financial derivatives, risk management, and other applications in quantitative finance.
Overall, infinite series are a powerful mathematical tool with diverse applications across various disciplines.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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