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The limit of a sequence is essentially the value that the terms of a sequence approach as the index (often denoted as \(n\)) becomes very large. In mathematical terms, for a sequence \(a_n\), the limit is denoted as \(\lim_{{n \to \infty}} a_n = L\). This means that the terms of the sequence get closer and closer to a number \(L\) as \(n\) increases.

For the given sequence \(a_n = 1 + \left(-\frac{1}{2}\right)^n\), observing the behavior of the sequence as \(n\) becomes larger can help us identify the limit. As \(n\) grows, the component \(\left(-\frac{1}{2}\right)^n\) gets smaller and tends to zero, since the base being a fraction less than 1 (in absolute value) means successive powers get closer to zero. Thus, \(a_n\) approaches \(1\) because \(\lim_{{n \to \infty}} 1 + \left(-\frac{1}{2}\right)^n = 1\).

• A sequence converges if it has a limit.
• In this problem, the sequence converges to \(1\).

Graphing sequences involves plotting the sequence terms \((n, a_n)\) on a coordinate grid where \(n\), the index, is on the x-axis, and \(a_n\), the sequence term's value, is on the y-axis. This provides a visual representation of the behavior of the sequence over increasing values of \(n\).

In this exercise, each term calculated was plotted from \(n = 1\) to \(n = 10\). When you connect the dots or just observe the plotted points, you can see the pattern or trend of the sequence. The graph can show whether the sequence is converging, diverging, oscillating, or constant.

For \(a_n = 1 + \left(-\frac{1}{2}\right)^n\), the graph shows an oscillating pattern around \(y = 1\). Over time, as the sequence's terms were plotted, their deviations from the line \(y = 1\) became smaller, indicating convergence towards the line.

To calculate terms of a sequence, usually, a formula for the general term \(a_n\) is given. By substituting different values of \(n\) into this formula, you can find specific terms of the sequence.

In this sequence, \(a_n = 1 + \left(-\frac{1}{2}\right)^n\), individual terms were calculated by plugging \(n = 1, 2, \ldots, 10\) into the formula:

• \(a_1 = 0.5\)
• \(a_2 = 1.25\)
• \(a_3 = 0.875\)
• \(a_4 = 1.0625\)
• \(a_5 = 0.9688\)
• \(a_6 = 1.0156\)
• \(a_7 = 0.9922\)
• \(a_8 = 1.0039\)
• \(a_9 = 0.9980\)
• \(a_{10} = 1.00098\)

The result was an alternating sequence that appears to stabilize to a specific value as \(n\) increases.

Accurately calculating these terms is crucial for correctly predicting the sequence's behavior, which aids in graphing and examining convergence.

A convergent sequence is one where its terms approach a specific value, known as the limit, as \(n\) (the term number) becomes very large. This behavior is indicative of a sequence 'settling down' into a particular value and not increasing or decreasing indefinitely.

For the exercise sequence \(a_n = 1 + \left(-\frac{1}{2}\right)^n\), each term was calculated, and it was observed that the terms increasingly stayed closer to \(1\) as \(n\) grew larger.

The sequence alternates above and below the line \(y = 1\) due to the alternating sign of \(\left(-\frac{1}{2}\right)^n\), but these fluctuations decreased in size with increasing \(n\).

This suggests convergence, as the sequence does not diverge or oscillate widely about a non-limiting value. Hence, we conclude that this sequence is convergent with the limit \(L = 1\).