Does this infinite series make sense ?

In summary, the given series g(x) is defined as the sum of a_n divided by the square root of x-n, where the coefficients a_n are real numbers. However, since the series is summed over all positive integers, for any real value of x, the term (x-n) will be negative and result in a complex function. This can also be approximated by an integral, but the convergence may vary depending on the behavior of the coefficients a_n.
  • #1
mhill
189
1
given the series

[tex] g(x)= \sum_{n=0}^{\infty}\frac{a_{n}}{\sqrt {x-n}} [/tex]

where the coefficients a_n are real numbers my question is does the above makes sense ? i mean since we are summing over all positive integers , no matter how big we choose 'x' there will be a factor so x-n n=0,1,2,3,4,..... (x-n) <0 then g(x) is complex no matter what real x we put.

If we approximate the series by an integral [tex] \int_{0}^{\infty}dr a(r) (x-r)^{-1/2} [/tex] we get a similar result, i do not know if this is paradoxical and i should have taken the absolute value |x| inside the square root.
 
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  • #2
It makes sense mathematically. You end up with a complex function. Convergence may be another matter - depending on the behavior of an.
 

1. What is an infinite series?

An infinite series is a sum of an infinite number of terms. Each term in the series is added together to form the total sum.

2. How do you know if an infinite series makes sense?

An infinite series makes sense if the terms in the series approach zero as the number of terms increases, meaning that the series converges to a finite value. This can be determined using various mathematical tests such as the comparison test, ratio test, or integral test.

3. What happens if an infinite series does not make sense?

If an infinite series does not make sense, it means that the terms in the series do not approach zero as the number of terms increases, and the series diverges to either positive or negative infinity. In this case, the sum of the series is undefined.

4. Can an infinite series make sense but not converge?

Yes, an infinite series can make sense but not converge. This means that the terms in the series approach zero, but the series does not reach a finite value. In this case, the series is said to diverge to infinity or oscillate between positive and negative values.

5. How is an infinite series used in science?

Infinite series are used in various areas of science, such as physics, engineering, and economics, to model real-world phenomena. They are particularly useful in calculating and predicting values that cannot be measured directly, such as the position of a moving object or the behavior of a complex system.

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