- #1
Bashyboy
- 1,421
- 5
I am having a difficult time seeing how [itex]\sum_{n=0}^{\infty} ((-1)^n + 1)x^n[/itex] is equivalent to [itex]2\sum_{n=0}^{\infty} x^{2n}[/itex]
It is not "trivial" it is zero. Zero terms do not change the sum so it does not matter if you regard them them or not.Oh, I see. So, any odd power would give a trivial answer, and we would disregard those?
The equivalence of two infinite series means that the two series have the same sum, or in other words, they converge to the same value. This means that as the number of terms in the series approaches infinity, the sum of the terms in both series will approach the same value.
The equivalence of two infinite series can be proven using various methods such as the comparison test, the limit comparison test, or the ratio test. These tests involve comparing the terms of the series to known series or using the properties of limits to determine the convergence of the series.
Yes, two infinite series with different terms can be equivalent. The terms of the series may be different, but if they have the same sum or converge to the same value, then the series can be considered equivalent.
Proving the equivalence of two infinite series is important in mathematics as it helps to determine the convergence or divergence of a series. It also allows us to manipulate and simplify complex series by replacing them with equivalent series that are easier to work with.
Yes, the equivalence of two infinite series has many real-world applications in fields such as engineering, physics, and economics. One example is in the calculation of compound interest, where the sum of an infinite series is used to determine the value of an investment over time.