How do you determine if #a_n=(11/8)+(1/81/27)+(1/271/64)+...+(1/n^31/(n+1)^3)+...# converge and find the sums when they exist?
The sum of the first
# S_n = 1  1/(n+1)^3 #
The sum to infinity is given by:
# S = 1 #
This is a difference sum, and we can see that almost all terms will cancel with other terms:
After which we are left with:
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To determine if the series ( a_n = (1  \frac{1}{8}) + (\frac{1}{8}  \frac{1}{27}) + (\frac{1}{27}  \frac{1}{64}) + \ldots + (\frac{1}{n^3}  \frac{1}{(n+1)^3}) + \ldots ) converges and find the sums when they exist, we can analyze the series as a telescoping series.

First, let's rewrite the terms of the series: [ a_n = 1  \frac{1}{8} + \frac{1}{8}  \frac{1}{27} + \frac{1}{27}  \frac{1}{64} + \ldots + \frac{1}{n^3}  \frac{1}{(n+1)^3} + \ldots ]

Notice that each term cancels out with the next term, except for the first and last terms: [ a_n = 1  \frac{1}{(n+1)^3} ]

Now, we can rewrite the series as: [ a_n = 1  \frac{1}{(n+1)^3} ]

To find out if the series converges, we need to take the limit of ( a_n ) as ( n ) approaches infinity: [ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(1  \frac{1}{(n+1)^3}\right) ]

Taking the limit, we get: [ \lim_{n \to \infty} \left(1  \frac{1}{(n+1)^3}\right) = 1  \lim_{n \to \infty} \frac{1}{(n+1)^3} = 1  0 = 1 ]

Since the limit of ( a_n ) is a finite number (1), the series converges.

To find the sum of the series, we sum up all the terms: [ S = \sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} \left(1  \frac{1}{(n+1)^3}\right) ]

This is a finite series, and we can find its sum using the formula for the sum of a finite geometric series: [ S = \frac{a_1}{1  r} ] where ( a_1 ) is the first term and ( r ) is the common ratio.

In this series, ( a_1 = 1 ) and ( r = 0 ) (since each term cancels out with the next term).

Substituting the values into the formula, we get: [ S = \frac{1}{1  0} = \frac{1}{1} = 1 ]
Thus, the sum of the series is 1.
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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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