Simon Bridge said:What happens when x=-1 or |x|>1?
i.e. compare the series with what it is supposed to represent.
note: http://hyperphysics.phy-astr.gsu.edu/hbase/math/lnseries.html
The power series you refer to is supposed to represent ln(x+1) ... so you have noticed that ln(x+1) is only defined for x>-1. This should tell you part of the answer to your question....what does it represent?
I can but I won't - that is perilously close to doing work for you, that you are best advised to do yourself....can you show me how it can converges please?
A power series is an infinite series of the form ∑n=0∞ an(x-c)n, where an are constants and c is a fixed number. It is a way of representing a function as a sum of terms with increasing powers of x.
Power series are useful for approximating functions that are difficult to evaluate directly. By truncating the series at a finite number of terms, we can get an approximation of the original function that is often accurate enough for practical purposes.
The convergence of a power series depends on the value of the variable x. A power series will converge for all values of x within a certain interval called the interval of convergence. This interval can be determined using the ratio test or the root test.
No, not every function can be represented by a power series. The function must have a Taylor series expansion in order to be represented as a power series. Additionally, the interval of convergence for the power series may not include the entire domain of the function, so it may only be a valid approximation within a certain range of values.
Power series are used in a variety of applications, including physics, engineering, and economics. They are particularly useful for approximating complex functions in order to make calculations more manageable. For example, they are used in electrical engineering to model alternating currents, and in economics to study inflation and interest rates.