Finding Sums of Infinite Series - Math

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Question

Find the value for $\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$$

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Answer

To best understand, let's write out the series. So

$\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$ = $1+1/2+(1/2)^2$ + $(1/2)^3$+...+(1/2)^\infty$

We can see this is an infinite geometric series with each successive term being multiplied by $\frac{1}{2}$ .

A definition you may wish to remember is

where stands for the common ratio between the numbers, which in this case is $\frac{1}{2}$ or . So we get

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Question

Evaluate:

$1 - $\frac{1}{6}$ + $\frac{1}{36}$ - $\frac{1}{216}$ +...$

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Answer

This is a geometric series whose first term is and whose common ratio is $r = - $\frac{1}{6}$$ . The sum of this series is:

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Question

Evaluate:

$1 + $\frac{1}{7}$ + $\frac{1}{49}$ + $\frac{1}{343}$ +...$

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Answer

This is a geometric series whose first term is and whose common ratio is $r = $\frac{1}{7}$$ . The sum of this series is:

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Question

Find the value for $\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$$

Tap to reveal answer

Answer

To best understand, let's write out the series. So

$\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$ = $1+1/2+(1/2)^2$ + $(1/2)^3$+...+(1/2)^\infty$

We can see this is an infinite geometric series with each successive term being multiplied by $\frac{1}{2}$ .

A definition you may wish to remember is

where stands for the common ratio between the numbers, which in this case is $\frac{1}{2}$ or . So we get

← Didn't Know|Knew It →

Question

Evaluate:

$1 - $\frac{1}{6}$ + $\frac{1}{36}$ - $\frac{1}{216}$ +...$

Tap to reveal answer

Answer

This is a geometric series whose first term is and whose common ratio is $r = - $\frac{1}{6}$$ . The sum of this series is:

← Didn't Know|Knew It →

Question

Evaluate:

$1 + $\frac{1}{7}$ + $\frac{1}{49}$ + $\frac{1}{343}$ +...$

Tap to reveal answer

Answer

This is a geometric series whose first term is and whose common ratio is $r = $\frac{1}{7}$$ . The sum of this series is:

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Question

Find the value for $\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$$

Tap to reveal answer

Answer

To best understand, let's write out the series. So

$\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$ = $1+1/2+(1/2)^2$ + $(1/2)^3$+...+(1/2)^\infty$

We can see this is an infinite geometric series with each successive term being multiplied by $\frac{1}{2}$ .

A definition you may wish to remember is

where stands for the common ratio between the numbers, which in this case is $\frac{1}{2}$ or . So we get

← Didn't Know|Knew It →

Question

Evaluate:

$1 - $\frac{1}{6}$ + $\frac{1}{36}$ - $\frac{1}{216}$ +...$

Tap to reveal answer

Answer

This is a geometric series whose first term is and whose common ratio is $r = - $\frac{1}{6}$$ . The sum of this series is:

← Didn't Know|Knew It →

Question

Evaluate:

$1 + $\frac{1}{7}$ + $\frac{1}{49}$ + $\frac{1}{343}$ +...$

Tap to reveal answer

Answer

This is a geometric series whose first term is and whose common ratio is $r = $\frac{1}{7}$$ . The sum of this series is:

← Didn't Know|Knew It →

Question

Find the value for $\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$$

Tap to reveal answer

Answer

To best understand, let's write out the series. So

$\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$ = $1+1/2+(1/2)^2$ + $(1/2)^3$+...+(1/2)^\infty$

We can see this is an infinite geometric series with each successive term being multiplied by $\frac{1}{2}$ .

A definition you may wish to remember is

where stands for the common ratio between the numbers, which in this case is $\frac{1}{2}$ or . So we get

← Didn't Know|Knew It →

Question

Evaluate:

$1 - $\frac{1}{6}$ + $\frac{1}{36}$ - $\frac{1}{216}$ +...$

Tap to reveal answer

Answer

This is a geometric series whose first term is and whose common ratio is $r = - $\frac{1}{6}$$ . The sum of this series is:

← Didn't Know|Knew It →

Question

Evaluate:

$1 + $\frac{1}{7}$ + $\frac{1}{49}$ + $\frac{1}{343}$ +...$

Tap to reveal answer

Answer

This is a geometric series whose first term is and whose common ratio is $r = $\frac{1}{7}$$ . The sum of this series is:

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Knew It

Finding Sums of Infinite Series - Math

Find the value for

To best understand, let's write out the series. So We can see this is an infinite geometric series with each successive term being multiplied by . A definition you may wish to remember is where stands for the common ratio between the numbers, which in this case is or . So we get

Evaluate:

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

Evaluate:

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

Find the value for

To best understand, let's write out the series. So We can see this is an infinite geometric series with each successive term being multiplied by . A definition you may wish to remember is where stands for the common ratio between the numbers, which in this case is or . So we get

Evaluate:

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

Evaluate:

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

Find the value for

To best understand, let's write out the series. So We can see this is an infinite geometric series with each successive term being multiplied by . A definition you may wish to remember is where stands for the common ratio between the numbers, which in this case is or . So we get

Evaluate:

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

Evaluate:

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

Find the value for

To best understand, let's write out the series. So We can see this is an infinite geometric series with each successive term being multiplied by . A definition you may wish to remember is where stands for the common ratio between the numbers, which in this case is or . So we get

Evaluate:

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

Evaluate:

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

Find the value for $\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$$

Evaluate:

$1 - $\frac{1}{6}$ + $\frac{1}{36}$ - $\frac{1}{216}$ +...$

This is a geometric series whose first term is and whose common ratio is $r = - $\frac{1}{6}$$ . The sum of this series is:

Evaluate:

$1 + $\frac{1}{7}$ + $\frac{1}{49}$ + $\frac{1}{343}$ +...$

This is a geometric series whose first term is and whose common ratio is $r = $\frac{1}{7}$$ . The sum of this series is:

Find the value for $\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$$

Evaluate:

$1 - $\frac{1}{6}$ + $\frac{1}{36}$ - $\frac{1}{216}$ +...$

This is a geometric series whose first term is and whose common ratio is $r = - $\frac{1}{6}$$ . The sum of this series is:

Evaluate:

$1 + $\frac{1}{7}$ + $\frac{1}{49}$ + $\frac{1}{343}$ +...$

This is a geometric series whose first term is and whose common ratio is $r = $\frac{1}{7}$$ . The sum of this series is:

Find the value for $\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$$

Evaluate:

$1 - $\frac{1}{6}$ + $\frac{1}{36}$ - $\frac{1}{216}$ +...$

This is a geometric series whose first term is and whose common ratio is $r = - $\frac{1}{6}$$ . The sum of this series is:

Evaluate:

$1 + $\frac{1}{7}$ + $\frac{1}{49}$ + $\frac{1}{343}$ +...$

This is a geometric series whose first term is and whose common ratio is $r = $\frac{1}{7}$$ . The sum of this series is:

Find the value for $\sum_${i=0}^{\infty}$ $($\frac{1}{2}$)^i$$

Evaluate:

$1 - $\frac{1}{6}$ + $\frac{1}{36}$ - $\frac{1}{216}$ +...$

This is a geometric series whose first term is and whose common ratio is $r = - $\frac{1}{6}$$ . The sum of this series is:

Evaluate:

$1 + $\frac{1}{7}$ + $\frac{1}{49}$ + $\frac{1}{343}$ +...$

This is a geometric series whose first term is and whose common ratio is $r = $\frac{1}{7}$$ . The sum of this series is: