# Members' Papers

## Various Speakers

This annual event gives our members the chance to shine by presenting their own papers. The usual time is divided into a number of short lectures by members of the Invariants society. All members are welcome to give a paper!

### Phil Tootil, Transcendental Number Theory

It is commonly known that pi and e are irrational numbers, but there is more that can be said. In fact, they are transcendental, meaning that they satisfy no rational polynomial. There are many open problems about transcendental numbers, such as whether the number pi+e is transcendental or not. We shall investigate Schanuel’s Conjecture, an unproved result which provides answers to many of these.

### Sam Evington, The “Paradox” of conditional convergence

One of the most surprising results in elementary analysis is that re-arranging the order of the terms in an infinite series can change the sum. Indeed a theorem of Riemann states that for certain real series you can, by rearranging the terms, make it sum to any number you like! We shall investigate this phenomenon in the simplest cases and then study its generalisation in the setting of Banach Spaces.