Sums of Infinite Series - Pre-Calculus
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Find the value for 
Find the value for
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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
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
Evaluate:
Evaluate:
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This is a geometric series whose first term is 
and whose common ratio is 
. The sum of this series is:
This is a geometric series whose first term is
Evaluate:
Evaluate:
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This is a geometric series whose first term is 
and whose common ratio is 
. The sum of this series is:
This is a geometric series whose first term is
What is the sum of the following infinite series?
What is the sum of the following infinite series?
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This series is not alternating - it is the mixture of two geometric series.
The first series has the positive terms.
The second series has the negative terms.
The sum of these values is 3.5.
This series is not alternating - it is the mixture of two geometric series.
The first series has the positive terms.
The second series has the negative terms.
The sum of these values is 3.5.
What is the sum of the alternating series below?
What is the sum of the alternating series below?
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The alternating series follows a geometric pattern.
We can evaluate the geometric series from the formula.
The alternating series follows a geometric pattern.
We can evaluate the geometric series from the formula.
Find the sum of the following infinite series:
Find the sum of the following infinite series:
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Notice that this is an infinite geometric series, with ratio of terms = 1/3. Hence it can be rewritten as:
Since the ratio, 1/3, has absolute value less than 1, we can find the sum using this formula:
Where 
is the first term of the sequence. In this case 
, and thus:
Notice that this is an infinite geometric series, with ratio of terms = 1/3. Hence it can be rewritten as:
Since the ratio, 1/3, has absolute value less than 1, we can find the sum using this formula:
Where
In the infinite series 
each term 
such that the first two terms are 
and 
. What is the sum of the first eight terms in the series?
In the infinite series 
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Once you're identified the pattern in the series, you might see a quick way to perform the summation. Since the base of the exponent for each term is negative, the result will be positive if 
is even, and negative if it is odd. And the series will just list the first 8 powers of 2, with that positive/negative rule attached. So you have:
-2, 4, -8, 16, -32, 64, -128, 256
Note that each "pair" of adjacent numbers has one negative and one positive. for the first pair, -2 + 4 = 2. For the second, -8 + 16 = 8. For the third, -32 + 64 = 32. And so for the fourth, -128 + 256 = 128. You can then quickly sum the values to see that the answer is 170.
Once you're identified the pattern in the series, you might see a quick way to perform the summation. Since the base of the exponent for each term is negative, the result will be positive if 
-2, 4, -8, 16, -32, 64, -128, 256
Note that each "pair" of adjacent numbers has one negative and one positive. for the first pair, -2 + 4 = 2. For the second, -8 + 16 = 8. For the third, -32 + 64 = 32. And so for the fourth, -128 + 256 = 128. You can then quickly sum the values to see that the answer is 170.
Find the value for
Find the value for
Tap to see back →
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
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
Evaluate:
Evaluate:
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This is a geometric series whose first term is and whose common ratio is . The sum of this series is:
This is a geometric series whose first term is
Evaluate:
Evaluate:
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This is a geometric series whose first term is and whose common ratio is . The sum of this series is:
This is a geometric series whose first term is
What is the sum of the following infinite series?
What is the sum of the following infinite series?
Tap to see back →
This series is not alternating - it is the mixture of two geometric series.
The first series has the positive terms.
The second series has the negative terms.
The sum of these values is 3.5.
This series is not alternating - it is the mixture of two geometric series.
The first series has the positive terms.
The second series has the negative terms.
The sum of these values is 3.5.
What is the sum of the alternating series below?
What is the sum of the alternating series below?
Tap to see back →
The alternating series follows a geometric pattern.
We can evaluate the geometric series from the formula.
The alternating series follows a geometric pattern.
We can evaluate the geometric series from the formula.
Find the sum of the following infinite series:
Find the sum of the following infinite series:
Tap to see back →
Notice that this is an infinite geometric series, with ratio of terms = 1/3. Hence it can be rewritten as:
Since the ratio, 1/3, has absolute value less than 1, we can find the sum using this formula:
Where is the first term of the sequence. In this case , and thus:
Notice that this is an infinite geometric series, with ratio of terms = 1/3. Hence it can be rewritten as:
Since the ratio, 1/3, has absolute value less than 1, we can find the sum using this formula:
Where
In the infinite series each term such that the first two terms are and . What is the sum of the first eight terms in the series?
In the infinite series
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Once you're identified the pattern in the series, you might see a quick way to perform the summation. Since the base of the exponent for each term is negative, the result will be positive if is even, and negative if it is odd. And the series will just list the first 8 powers of 2, with that positive/negative rule attached. So you have:
-2, 4, -8, 16, -32, 64, -128, 256
Note that each "pair" of adjacent numbers has one negative and one positive. for the first pair, -2 + 4 = 2. For the second, -8 + 16 = 8. For the third, -32 + 64 = 32. And so for the fourth, -128 + 256 = 128. You can then quickly sum the values to see that the answer is 170.
Once you're identified the pattern in the series, you might see a quick way to perform the summation. Since the base of the exponent for each term is negative, the result will be positive if
-2, 4, -8, 16, -32, 64, -128, 256
Note that each "pair" of adjacent numbers has one negative and one positive. for the first pair, -2 + 4 = 2. For the second, -8 + 16 = 8. For the third, -32 + 64 = 32. And so for the fourth, -128 + 256 = 128. You can then quickly sum the values to see that the answer is 170.
Find the value for
Find the value for
Tap to see back →
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
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
Evaluate:
Evaluate:
Tap to see back →
This is a geometric series whose first term is and whose common ratio is . The sum of this series is:
This is a geometric series whose first term is
Evaluate:
Evaluate:
Tap to see back →
This is a geometric series whose first term is and whose common ratio is . The sum of this series is:
This is a geometric series whose first term is
What is the sum of the following infinite series?
What is the sum of the following infinite series?
Tap to see back →
This series is not alternating - it is the mixture of two geometric series.
The first series has the positive terms.
The second series has the negative terms.
The sum of these values is 3.5.
This series is not alternating - it is the mixture of two geometric series.
The first series has the positive terms.
The second series has the negative terms.
The sum of these values is 3.5.
What is the sum of the alternating series below?
What is the sum of the alternating series below?
Tap to see back →
The alternating series follows a geometric pattern.
We can evaluate the geometric series from the formula.
The alternating series follows a geometric pattern.
We can evaluate the geometric series from the formula.
Find the sum of the following infinite series:
Find the sum of the following infinite series:
Tap to see back →
Notice that this is an infinite geometric series, with ratio of terms = 1/3. Hence it can be rewritten as:
Since the ratio, 1/3, has absolute value less than 1, we can find the sum using this formula:
Where is the first term of the sequence. In this case , and thus:
Notice that this is an infinite geometric series, with ratio of terms = 1/3. Hence it can be rewritten as:
Since the ratio, 1/3, has absolute value less than 1, we can find the sum using this formula:
Where