This is a bunch of prerequisites to understand section 3.5 of Ram P. Kanwal.

There are three properties listed here:

- is absolutely and uniformly convergent for all values of s and t in the circle
- It satisfies integrodifferential equations

## Prerequisites

### Property 1

**Direct Comparison Test**(Wikipedia): If and then is also absolutely convergent. (Make note of variants in the Wikipedia Page in NET Notebook.) Here,- Two Assumptions: 1. , 2. is a constant.
**Schwartz Inequality**(Wikipedia): Fot two square integrable functions,**Convergence of Geometric Series**(Wikipedia): If

### Property 2

In step 2, they are changing limits from m=2 to m=1. If it is confusing, you can try substituting p = m-1.

### Property 3

If a series is uniformly convergent, summation and integral can be interchanged (Proof – Math.SE).

## Some Extra Links

- Compilation of Series Convergence Tests (Wikipedia)
- Some important sequences and series (Blog)
- Generalisation of Schwartz Inequality: Hölder inequality (Wikipedia)
- Also worth reading about application of Schwartz Inequality – used in Analysis, Algebra, Statistics, and a variety of fields!
- A video discussing shifting limits of an infinite series (YouTube)

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