Is infinity a variable or a constant

Sum symbol

The summation sign (also called “summation sign” or “summation symbol”) is used when we want to form a sum over a finite or infinite number of terms in a series, with the value of the terms depending on a variable. A sum is written with the sum sign in this way:

The sum symbol includes various components, which are color-coded above:

  • The Run variable, or the Running index (blue): This is the changing variable of the sum. There is exactly one summand for each value of the run variable. The letters i, j, k or l elected.
  • The Starting value (red): This is the smallest value that the run variable assumes. The starting value is an integer.
  • The Final value (green): This is the greatest value that the run variable assumes. The final value is an integer. If there is the infinity symbol (a lying eight: ∞), the infinite sum is formed, whereby the variable assumes the value of the start value and any integer greater than the start value.
  • The Summands (orange): The summands usually consist of a function that depends on the run variable.

If the sum is written with start and end value as above, the run variable takes the value of every whole number between the start and end value (including start value and end value). Exactly one summand is added for each of its values. An example of such a sum is the sum of all the squares of the whole numbers between one and five:

The run variable is here i, the start value 1, the end value 5 and the summands are set using the function i2 formed, where i takes all values ​​from one to five.

Calculation rules for the summation sign

The sum symbol is an abbreviated form of sums. Therefore, we can remove individual summands from the summation sign and note them down individually:

The first step here was the summand an removed from the sum symbol and noted individually. The final value was therefore reduced by one. Then there was also the summand an-1 removed from the sum symbol. The final value was reduced again by one. In the last step, all summands were removed from the summand.

Of course, we can not only start from the end value in order to note down summands individually, but also from the start value. The start value must be adjusted accordingly:

We can also split a total into several partial totals:

In this way we can also individually note summands for running values ​​between the start and end value:

Sums over constant summands

Sometimes it happens that the individual summands do not depend on the value of the run variable at all. ai is then equal to a constant c. In these cases, we can simply calculate the sum by multiplying the value of the constant by the number of summands. The number of summands in a sum corresponds to the difference between the end value and the start value plus one:

For example, it looks like this:

Associative law

The associative law says that it does not matter in which order the summands of a sum are added up. We can make use of this if the summands themselves contain sums again:

Distributive law

The distributive law says that it does not matter whether you first multiply each summand by a factor and then form the sum of the products, or whether you first add the summands and then multiply the sum by the factor. Of course, this also applies to the sum symbol. So we can exclude factors from the summands and write them in front of the sum sign: