The corresponding infinite sum is related to the "incomplete theta function", $$\Theta_0(x,y)=\sum_{n\geq 0}x^ny^{n(n-1)/2}.$$ We have $$\sum_{n\geq 0}a^{n^2}=\Theta_0(a,a^2).$$ Accordingly, your series has a name "partial sum of the incomplete theta function".

Ref. A. Sokal, The leading root of the partial theta function, Adv. Math. 229 (2012), no. 5, 2603–2621,

and references there.

The corresponding infinite sum is related to the "incomplete theta function", $$\Theta_0(x,y)=\sum_{n\geq 0}x^ny^{n(n-1)/2}.$$ We have $$\sum_{n\geq 0}a^{n^2}=\Theta_0(a,a^2).$$ Accordingly, your series has a name "partial sum of the incomplete theta function".

Ref. A. Sokal, The leading root of the partial theta function, Adv. Math. 229 (2012), no. 5, 2603–2621,

and references there.

Such series were subject of a lot of research recently. On partial sums, see, for example

Katkova, Olga M.; Lobova, Tetyana; Vishnyakova, Anna M. On power series having sections with only real zeros. Comput. Methods Funct. Theory 3 (2003), no. 1-2, 425–441.