All High School Math Resources
Example Questions
Example Question #11 : Sequences And Series
Find the sum of all even integers from to .
The formula for the sum of an arithmetic series is
,
where is the number of terms in the series, is the first term, and is the last term.
We know that there are terms in the series. The first term is and the last term is . Our formula becomes:
Example Question #1 : Finding Partial Sums In A Series
Find the sum of all even integers from to .
The formula for the sum of an arithmetic series is
,
where is the number of terms in the series, is the first term, and is the last term.
Example Question #2063 : High School Math
Find the sum of the even integers from to .
The sum of even integers represents an arithmetic series.
The formula for the partial sum of an arithmetic series is
,
where is the first value in the series, is the number of terms, and is the difference between sequential terms.
Plugging in our values, we get:
Example Question #31 : Sequences And Series
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
Example Question #2 : Finding Sums Of Infinite Series
Evaluate:
The series does not converge.
This is a geometric series whose first term is and whose common ratio is . The sum of this series is:
Example Question #2 : Sums Of Infinite Series
Evaluate:
The series does not converge.
This is a geometric series whose first term is and whose common ratio is . The sum of this series is:
Example Question #1 : Limits
A limit describes what -value a function approaches as approaches a certain value (in this case, ). The easiest way to find what -value a function approaches is to substitute the -value into the equation.
Substituting for gives us an undefined value (which is NOT the same thing as 0). This means the function is not defined at that point. However, just because a function is undefined at a point doesn't mean it doesn't have a limit. The limit is simply whichever value the function is getting close to.
One method of finding the limit is to try and simplify the equation as much as possible:
As you can see, there are common factors between the numerator and the denominator that can be canceled out. (Remember, when you cancel out a factor from a rational equation, it means that the function has a hole -- an undefined point -- where that factor equals zero.)
After canceling out the common factors, we're left with:
Even though the domain of the original function is restricted ( cannot equal ), we can still substitute into this simplified equation to find the limit at
Example Question #1 : Finding One Sided Limits
Let .
Find .
The limit does not exist.
This is a graph of . We know that is undefined; therefore, there is no value for . But as we take a look at the graph, we can see that as approaches 0 from the left, approaches negative infinity.
This can be illustrated by thinking of small negative numbers.
NOTE: Pay attention to one-sided limit specifications, as it is easy to pick the wrong answer choice if you're not careful.
is actually infinity, not negative infinity.
Example Question #3 : Limits
Evaluate the limit below:
1
0
will approach when approaches , so will be of type as shown below:
So, we can apply the L’ Hospital's Rule:
since:
hence: