The rule for the sum of this series?

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In summary, the given series follows the pattern $$\sum_{n=0}^{\infty} \frac{1}{(4n+1)(4n+3)}$$ and can be evaluated as $$\frac{\pi}{8}$$.
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Saracen Rue
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What is the rule for the following series: $$\frac {1}{1×3}+\frac {1}{5×7}+\frac {1}{9×11}...$$
Consider the following series with the following pattern $$\frac {1}{1×3}+\frac {1}{5×7}+\frac {1}{9×11}...$$

How would you go about working out what the general rule for this sum is? That is in the form of ##\sum_{n=a}^{b}f(n)##

Any help is greatly appreciated.
 
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Your sum is $$\sum_{n=0}^\infty \frac{1}{(4n+1)(4n+3)}$$

Do you have to evaluate it?
 
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Math_QED said:
Your sum is $$\sum_{n=0}^\infty \frac{1}{(4n+1)(4n+3)}$$

Do you have to evaluate it?
Thank you so much, this had really been bugging me. I actually did try testing that mentally but I guess I messed something up.
 
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Math_QED said:
Your sum is $$\sum_{n=0}^\infty \frac{1}{(4n+1)(4n+3)}$$

Do you have to evaluate it?
Continuing the processing: [itex]\sum_{n=0}^{\infty}\frac{1}{(4n+1)(4n+3)}=\frac{1}{2}\sum_{n=0}^{\infty}(\frac{1}{4n+1}-\frac{1}{4n+3})=\frac{\pi}{8} [/itex]
 
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Related to The rule for the sum of this series?

What is the rule for the sum of this series?

The rule for the sum of a series is a mathematical formula used to determine the total value of a series or sequence of numbers. It is also known as the sum formula or summation formula.

How do you find the sum of a series?

To find the sum of a series, you can use the following formula: S = (n/2)(a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term. This formula works for both arithmetic and geometric series.

What is the difference between an arithmetic and geometric series?

An arithmetic series is a sequence of numbers where the difference between each term is constant. A geometric series is a sequence of numbers where the ratio between each term is constant. In other words, in an arithmetic series, you add a constant number to each term to get the next one, while in a geometric series, you multiply by a constant number to get the next term.

What is the formula for an arithmetic series?

The formula for an arithmetic series is: S = (n/2)(2a + (n-1)d), where S is the sum, n is the number of terms, a is the first term, and d is the common difference between each term.

What is the formula for a geometric series?

The formula for a geometric series is: S = a(1-r^n)/(1-r), where S is the sum, a is the first term, r is the common ratio between each term, and n is the number of terms.

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