Question

Prove that a series of complex terms diverges if \(\rho>1(\rho=\) ratio test limit). Hint: The \(n\) th term of a convergent series tends to zero.

Short answer

A series diverges if \( \rho > 1 \) since the terms do not tend to zero.

Step-by-step solution

- Understand the Ratio Test

The ratio test involves the limit \[ \rho = \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \] where \(a_n\) are the terms of the series.

- Identify the Series' Ratio

Identify the ratio limit \( \rho \) of the given series. According to the problem, \( \rho > 1 \).

- Apply the Ratio Test Conclusion

Based on the ratio test, if \( \rho > 1 \), the series \( \sum a_n \) diverges. This is because the terms do not tend to zero and/or may increase in magnitude.

- State the Divergence Condition

State that for any series to be convergent, the \( n \)-th term must tend to zero as \( n \to \infty \). Since here \( \rho > 1 \), the terms often grow larger.

- Conclude Divergence

Since the terms of the series do not tend to zero and the ratio \( \rho \) is greater than one, conclude that the series diverges.

Key concepts

complex series divergence

To understand why a series of complex terms diverges when \(\rho > 1\), we need to delve into what divergence truly means.
Divergence essentially indicates that the sum of the terms in the series doesn't settle to a finite number.
In terms of a complex series, which may contain real and imaginary parts, this divergence implies that both parts fail to converge.

When we apply the ratio test to a series, we examine the limit: \rho = \lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{{a_n}} \right|\
Here, \({a_n}\) represents the terms of the series.

If the ratio test reveals that \(\rho > 1\), it suggests that the subsequent terms grow compared to the previous terms.
This growth prevents the series' sum from stabilizing, leading to divergence.

In simpler terms, since the terms of such a series increase or oscillate without settling down, the series can't sum to a finite value and hence diverges.

ratio test limit

The ratio test is a standard tool used to determine whether a series converges or diverges.
We begin by looking at the limit: \rho = \lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{{a_n}} \right|\
where \({a_n}\) is the sequence of terms in the series.

If:

• \(\rho < 1\) the series converges.


• \(\rho > 1\) the series diverges.


• \(\rho = 1\), the ratio test is inconclusive.

Let's consider if \(\rho > 1\).
This condition shows that each successive term is larger than the previous term.
The terms of the series grow and grow, indicating that the series sum is unbounded.
Therefore, the series diverges.
Calculating the limit often involves algebraic manipulation and simplification but always targets whether the value of the ratio exceeds one.

terms tending to zero

For a series to converge, one of the most fundamental requirements is that the terms of the series must tend to zero.
That is, as \(n \to \infty\), \(a_n \to 0\).
If this doesn't happen, the series can't possibly converge.

Consider the given exercise.
If \(\rho > 1\), it tells us that the terms do not shrink; instead, they often grow larger.
Therefore, \( \left| a_{n+1} \right| > \left| a_n \right| \) as n increases.
Consequently, the terms don't tend toward zero.

When the terms' magnitudes increase or fail to decrease, the sum of these terms will not settle.
This is central to many convergence tests, the ratio test included.
When proving divergence, always ensure to show that the terms do not approach zero under the given ratio conditions.