Which geometric series converges?
1 answer:
Based on <em>ratio</em> criterion and definition of <em>geometric</em> series we conclude that the<em> geometric</em> series 1/5 + 1/20 + 1/80 + 1/320 + ... is convergent.
<h3>How to determine the convergence of a geometric series</h3>
<em>Geometric</em> series are <em>discrete</em> formulas of the form , where a and r are real numbers, and n and b are <em>natural</em> numbers. According to the <em>ratio</em> criterion, a <em>geometric</em> series is convergent if and only if |r| ≤ 1.
In accordance to this information, we conclude that the geometric series 1/5 + 1/20 + 1/80 + 1/320 + ... is convergent.
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Answer:
256
Step-by-step explanation:
A calculator works well for this.
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None of the minus signs are subject to the exponents (because they are not in parentheses, as (-1)^5, for example. Since there are an even number of them in the product, their product is +1 and they can be ignored.
1 to any power is still 1, so the factors (1^n) can be ignored.
After you ignore all of the things that can be ignored, your problem simplifies to ...
(2^2)(2^-3)^-2
The rules of exponents applicable to this are ...
(a^b)^c = a^(b·c)
(a^b)(a^c) = a^(b+c)
Then your product simplifies to ...
(2^2)(2^((-3)(-2)) = (2^2)(2^6)
= 2^(2+6)
= 2^8 = 256
