# List of Notes - Category: Sequence Theorems

## Squeeze (Sandwich) Theorem for Sequences

Consider two sequences x_n and y_n. When we write that x_n=y_n we mean that corresponding values are equal, i.e. x_1=y_1, x_2=y_2 etc.

Fact 1. If two sequences x_n and y_n are equal: x_n=y_n, and each of them has limit (finite or infinite): lim x_n=a and lim y_n=b then a=b.

## Algebra of Limit of Sequence

Consider two sequences x_n and y_n. When we talk about sum of these sequences, we talk about sequence x_n+y_n, whose elements are x_1+y_1,x_2+y_2,x_3+y_3..... Same can be said about other arithmetic operations. In other words sum of sequences is sequence with elements that are sum of corresponding elements of initial two sequences.

## Indeterminate Form for Sequence

When we described arithmetic operations on limits, we made assumption that sequences approach finite limits.

Now, let's consider case when limits are infinite or, in the case of quotient, limit of denominator equals 0.

## Stolz Theorem

To find limits of indeterminate expressions (x_n)/(y_n) of type (oo)/(oo) often can be useful following theorem.

Stolz Theorem. Suppose that sequence y_n->+oo and starting from some number with increasing of n also increases y_n (in other words if m>n then y_m>y_n). Then lim (x_n)/(y_n)=lim (x_n-x_(n-1))/(y_n-y_(n-1)) if limit of expression on the right side exists (finite or infinite).