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Mathematical Expression Editor
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Remember our facts about factorials.
The Ratio Test is a useful convergence test that is very useful when trying to
determine convergence or divergence of series whose summands involve products of
polynomials, factorials, and exponential functions.
A fundamental skill when using this test is to be able to simplify ratios of expressions
involving these types of terms.
This assignment will review some of the algebraic skills required when using this test.
Ratios of exponentials
Ratios of exponential terms also simplify nicely. Suppose and . Simplify:
Suppose is a sequence whose th term is given by: Note: The is in the exponent.
Simplify
Factorials
Given an integer , the notation “” is defined as follows: For instance, . Compute:
How did you compute in the last problem? You could certainly find
and separately, then divide, but there is a nicer way to do this:
Ratios of factorials are always easiest to compute by canceling like terms! Compute
the following, simplify your final answers.
Often, factorials arise in sequences, and it is important to understand how to
write out various terms. Suppose is a sequence whose th term is given by:
(a)
(b)
Find an expression for . Express your answer in the form .
(c)
Calculate and simplify:
Let be a sequence whose th term is given by: Calculate and simplify:
Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)
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Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)