Hint: Use the binomial series \(\displaystyle (1+x)^\alpha = \sum_{k=0}^\infty {\alpha\choose k}x^k,\) with $x = -(1-p)$ and $\alpha = 1-r.$pp123123 said:I came across some summation but have no idea how to simplify it.
$\sum_{x=0}^{\infty} \binom{x+r-2}{r-2}(1-p)^{x}=p^{1-r}$
Why is it so?
Summation is a mathematical operation that involves adding a series of numbers together. It is important in science because it allows us to calculate total values, such as the total amount of a certain substance or the total energy in a system.
To perform summation using sigma notation, you need to first identify the starting and ending values of the series. Then, you can write the sum of all the terms in the series using the sigma symbol (∑). For example, if you want to calculate the sum of the first 10 natural numbers, the sigma notation would be ∑n, where n=1 to 10.
Yes, summation can be used for both finite and infinite series. Finite series have a defined starting and ending point, while infinite series continue indefinitely. In both cases, the sigma notation can be used to represent the sum of all the terms in the series.
Some common properties of summation include the commutative property (changing the order of terms does not affect the sum), the associative property (grouping terms does not affect the sum), and the distributive property (distributing a factor to all terms in the series).
Summation is used in various real-world scientific applications, such as calculating the total mass of a compound in chemistry, determining the total energy in a circuit in physics, and estimating the total population of a species in biology. It is a useful tool for analyzing and understanding complex systems and data sets.