Calculate, to four decimal places, the first ten terms of...

Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand.
Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
$a_n=1+\left(-\frac{1}{2}\right)^n$

Key Concepts

Sequence

A sequence is an ordered list of numbers defined by a rule that determines the value of each term based on its position. In mathematical analysis, sequences are studied to understand their behavior as the term number increases, and many sequences have properties such as eventual convergence to a limit or divergence.

Geometric Sequence

A geometric sequence is one in which each term is obtained by multiplying the previous term by a fixed constant called the common ratio. Even when the defining formula includes an additive constant, as in some modified sequences, the exponential term often behaves like a geometric sequence, which is key to determining the overall behavior of the sequence.

Alternating Sequence

An alternating sequence is a sequence in which the terms alternate in sign. This behavior typically arises when the common ratio of an associated geometric sequence is negative. Alternating sequences can converge if the magnitude of the terms decreases sufficiently rapidly, despite the change in sign.

Convergence and Limit of a Sequence

The concept of convergence in sequences refers to the idea that as the number of terms increases indefinitely, the sequence approaches a specific value, known as the limit. Determining the limit, if it exists, involves analyzing the behavior of the sequence's general term as the term number tends toward infinity.

Graphical Representation of Sequences

Graphing a sequence involves plotting discrete points corresponding to the terms of the sequence. This visual representation helps in understanding the behavior of the sequence, such as whether it approaches a limiting value, oscillates, or diverges, and offers insight into the nature of the sequence’s convergence or divergence.

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