- #1
phonic
- 28
- 0
Is it possible to get a analytical result for this series? It looks simple:
[itex]\sum_{k=1} ^t a^{t-k}b^{k-1} [/itex]
Thanks a lot!
[itex]\sum_{k=1} ^t a^{t-k}b^{k-1} [/itex]
Thanks a lot!
The sum of a series can be calculated by adding all of the terms in the series together. This can be done manually or by using a formula such as the arithmetic or geometric series formula.
An arithmetic series has a constant difference between each term, while a geometric series has a constant ratio between each term. This means that in an arithmetic series, each term is added or subtracted by a fixed number, while in a geometric series, each term is multiplied or divided by a fixed number.
The nth term in a series can be found by using the formula: an = a1 + (n-1)d for an arithmetic series, or an = a1*r^(n-1) for a geometric series. Here, an represents the nth term, a1 is the first term, d is the common difference in an arithmetic series, and r is the common ratio in a geometric series.
The limit of a series is the value that the sum of the series approaches as the number of terms in the series goes to infinity. This is also known as the convergence of a series. If the limit of a series is a finite number, the series is said to converge, while if the limit is infinity or does not exist, the series is said to diverge.
There are multiple tests that can be used to determine if a series is convergent or divergent, such as the ratio test, the root test, and the integral test. These tests involve analyzing the behavior of the terms in the series and can help determine if the series will approach a finite limit or not.