1. Define what is meant by the limit of a series.
Please try to explain in words more than symbols.
The limit is the result you would get by adding up an infinite number of terms of the series. The larger the number of terms that you add, the closer you get to that limit.
To understand the concept of a series, let's take an example. Consider the series 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... In this series, each term is obtained by dividing the previous term by 2.
The limit of a series is the value that the terms of the series approach as we add more and more terms. In the case of the above series, if we were to keep adding more and more terms, we would notice that the sum gets closer and closer to the value of 2.
For instance, if we add the first few terms of the series, we get:
1 + 1/2 + 1/4 = 1.75
If we add more terms, such as the first five terms:
1 + 1/2 + 1/4 + 1/8 + 1/16 = 1.9375
And if we keep adding even more terms, the sum continues to get closer to 2:
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 1.96875
As we add an infinite number of terms, the sum of the series approaches the limit of 2. However, it is important to note that in most cases, we cannot actually reach the exact limit when dealing with an infinite series, as we can only approximate it using a finite number of terms. Nonetheless, we can mathematically prove that the limit exists and determine its value.