In this talk, we will learn about recursive formulas for Eisenstein series, some of which are classical, and some of which are surprisingly new. In particular, we will see that these important examples of modular forms can be recursively defined in many ways, which directly yields surprising identities between convolution sums of sums of divisor functions as well as relations among the classical Bernoulli numbers. Along the way, we will learn about important examples of doubly periodic, meromorphic functions, also known as elliptic functions, and their connections to modular forms. This talk will be self-contained, and no prior knowledge of modular forms or the related objects mentioned above will be assumed.