Accumulator Function

The accumulator function refers to a mathematical function that accumulates or sums up a sequence of values. In the context of this question, we are considering the accumulator function as n tends to infinity.

Stability of a Function

In control theory, the stability of a function refers to the behavior of the function over time. A stable function is one where any bounded input to the function produces a bounded output. Stability is a crucial property in control systems as it ensures that the system's response does not grow uncontrollably.

Stability Analysis

To determine the stability of the accumulator function as n tends to infinity, we need to analyze its behavior. Let's consider the accumulator function:

A(n) = A(n-1) + n

where A(n) represents the accumulator function at time n.

Stability Analysis of the Accumulator Function

When n tends to infinity, let's analyze the behavior of the accumulator function.

A(n) = A(n-1) + n

Substituting n-1 for n in the equation, we get:

A(n-1) = A(n-2) + n-1

Substituting the above equation into the original equation, we get:

A(n) = (A(n-2) + n-1) + n

Expanding the equation further, we have:

A(n) = A(n-2) + n-1 + n

Continuing this process, we can express A(n) as:

A(n) = A(n-k) + (n-1) + n + (n+1) + ... + (n-k+1)

Simplifying the equation, we get:

A(n) = A(n-k) + kn - (1 + 2 + ... + k-1 + k)

The term (1 + 2 + ... + k-1 + k) represents the sum of the first k natural numbers and can be expressed as k(k+1)/2.

Therefore, the equation becomes:

A(n) = A(n-k) + kn - k(k+1)/2

Conclusion

From the above analysis, we can observe that as n tends to infinity, the accumulator function A(n) grows without bound. This implies that for any bounded input, the output of the accumulator function is unbounded, which violates the definition of stability.

Hence, the correct answer is option 'C': The function is unstable.