Write the following in symbols :
The necessary and sufficient condition for an infinite series `sum u_n` to be convergent'is that limit of `u_n` as n tending to infinity must be zero.

Answer

To express the given statement "The necessary and sufficient condition for an infinite series \( \sum u_n \) to be convergent is that the limit of \( u_n \) as \( n \) tends to infinity must be zero" in symbolic form, we can follow these steps:

### Step 1: Identify the sub-statements
1. **Sub-statement 1 (p)**: "An infinite series \( \sum u_n \) is convergent."
2. **Sub-statement 2 (q)**: "The limit of \( u_n \) as \( n \) tends to infinity is 0."

### Step 2: Define the sub-statements in symbols
- Let \( p \) represent the statement "An infinite series \( \sum u_n \) is convergent."
- Let \( q \) represent the statement "The limit of \( u_n \) as \( n \) tends to infinity is 0."

### Step 3: Express the necessary and sufficient condition
In mathematical reasoning, the phrase "necessary and sufficient condition" indicates a bi-conditional relationship between the two statements. This means that both statements are equivalent; if one is true, the other must also be true.

### Step 4: Write the bi-conditional statement
The bi-conditional statement can be expressed as:
\[ p \iff q \]
This reads as "p if and only if q," meaning that \( p \) is true if \( q \) is true and vice versa.

### Final Answer
Thus, the symbolic representation of the given statement is:
\[ p \iff q \]
where:
- \( p \): "The infinite series \( \sum u_n \) is convergent."
- \( q \): "The limit of \( u_n \) as \( n \) tends to infinity is 0."

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