Examples of sum in the following topics:
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- $\sum_{i=1}^{n} y_{1}$
- $\sum_{i=1}^{n} x_{i}$
- $\sum_{i=1}^{n}x_{i}y_{i}-\frac{1}{n}\sum_{i=1}^{n}x_{i}\sum_{j=1}^{n}y_{j}$
- Calculate the denominator: The
sum of the squares of the $x$-coordinates minus one-eighth the sum of the $x$-coordinates squared.
- $\sum_{i=1}^{n}(x_{i}^{2})-\frac{1}{n}(\sum_{i=1}^{n}x_{i})^{2}$
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- An arithmetic sequence which is finite has a specific formula for its sum.
- We can come up with a formula for the sum of a finite arithmetic formula by looking at the sum in two different ways.
- First we think of it as the sum of terms that are written in terms of $a_1,$ so that the second term is $a_1+d,$ the third is $a_1+2d, \ldots.$ Then our sum looks like:
- If $a_1=d=0$, then the sum of the series is $0$.
- If either $a_1$ or $d$ is nonzero, then the infinite series has no sum.
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- Geometric series are one of the simplest examples of infinite series with finite sums.
- A geometric series with a finite sum is said to converge.
- What follows in an example of an infinite series with a finite sum.
- We will calculate the sum $s$ of the following series:
- Find the sum of the infinite geometric series $64+ 32 + 16 + 8 + \cdots$
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- Summation is the operation of adding a sequence of numbers, resulting in a sum or total.
- If numbers are added sequentially from left to right, any intermediate result is a partial sum.
- For finite sequences of such elements, summation always produces a well-defined sum.
- (This series sums to $126$.)
- Another example is $\displaystyle \sum_{i=3}^6 (i^2+1) = (3^2+1)+(4^2+1)+(5^2+1)+(6^2+1)=10+17+26+37.$ This series sums to $90.$ So we could write $\displaystyle \sum_{i=3}^6 (i^2+1)=90.$
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- By utilizing the common ratio and the first term of a geometric sequence, we can sum its terms.
- The sum of the terms also gets larger and larger, and the series has no sum.
- This is a different type of divergence and again the series has no sum.
- For $r\neq 1$, the sum of the first $n$ terms of a geometric series is:
- Calculate the sum of the first $n$ terms in a geometric sequence
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- If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation.
- For finite sequences of such elements, summation always produces a well-defined sum (possibly by virtue of the convention for empty sums).
- A series is merely the sum of the terms of a series.
- So the sum for the whole series is $\frac n2 (t_1+t_n)$ .
- As an example, let's find the sum $2+6+18+54+162$ .
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- For example, the sum of $2+3i$ and $5+6i$ can be calculated by add the two real parts $(2+5)$ and the two imaginary parts $(3+6)$ to produce the complex number $7+9i$.
- As another example, consider the sum of $1-3i$ and $4+2i$.
- Note that the same thing can be accomplished by imagining that you are distributing the subtraction sign over the sum $2+4i$ and then adding as defined above.
- Calculate the sums and differences of complex numbers by adding the real parts and the imaginary parts separately
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- It is the sum of a collection of numbers divided by the number of numbers in that collection.
- $\displaystyle A = \frac{1}{n} \sum_{i=1}^n a_i = \frac{1}{n}(a_1 + \cdots + a_n)$
- In order to find the average, we must first find the sum of the numbers:
- Next, divide their sum by 3, the number of values in the set:
- We need to add the values together and then divide that sum by the total number of values, which is 6.
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- Adding the second term to the second-to-last term also amounts to a sum of 20.
- The reason that the sum of the second pair equaled that of the first pair was that we went up by two on the left, and down by two on the right.
- As long as you go up by the same amount as you go down, the sum will stay the same—and this is just what happens for arithmetic series.
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- They have the form of a sum of scaled powers of a variable and are easy to study.
- The first and the second do not have a non-negative integer exponent and the third is a sum of two monomials.
- A polynomial over $\mathbb{R}$ is a finite sum of monomials over $\mathbb{R}$.
- is the finite sum of the $4$ monomials $4x^{13}, 3x^2, -\pi x$ and $1 = 1x^0.$ It is also the sum of the 6 mononomials
$1/3 x^{100}, -1/3 x^{100}, 4x^{13}, 3x^2, -\pi x$ and $1$, as will be explained in the discussion about addition and subtraction of polynomials.
- Every monomial is also a polynomial, as it can be written as a sum with one term, itself.