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The given image contains two mathematical problems.

### Problem 1.3a:
You are asked to compute the sum of the following series:

\[
S = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}
\]

Additionally, you are provided the error \(\Delta S = 10^{-3}\), meaning that your result for \(S\) should have an accuracy up to 3 decimal places.

### Solution for 1.3a:
Let's compute the sum of the series:

\[
S = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}
\]

I will now calculate the value of \(S\).The value of the sum \(S\) is approximately:

\[
S \approx 1.718
\]

### Problem 1.3b:
You are asked to compute the value of:

\[
S = \sqrt{30}
\]

with an uncertainty \(\delta_S = 0.1\%\).

### Solution for 1.3b:
We calculate the square root of 30, and then determine the uncertainty. The relative uncertainty is given as 0.1%, meaning:

\[
\delta_S = \frac{0.1}{100} = 0.001
\]

Now, I'll compute \(\sqrt{30}\) and apply the uncertainty.

Let me calculate this next.The value of \(S = \sqrt{30}\) is approximately:

\[
S \approx 5.477
\]

The uncertainty \(\delta_S\), given as 0.1%, is:

\[
\delta_S \approx 0.0055
\]

This means the value of \(S\) is \(5.477 \pm 0.0055\).

---

Would you like further details or explanations? Here are five related questions that build on this material:

1. How can we improve the accuracy of summing series like in 1.3a?
2. What is the significance of the error margin \(\Delta S = 10^{-3}\) in practical applications?
3. How do we propagate uncertainty in more complex calculations like powers or logarithms?
4. How does the choice of approximation method affect error bounds?
5. Can we derive a general formula for the harmonic series?

**Tip**: When working with square roots in calculations involving uncertainties, the relative error in the square root is half the relative error in the original value.

Tính giá trị các biểu thức sau đây: a) S = 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8, với ΔS = 10^-3. b) S = √30, với δS = 0.1%.

Learn how to solve a summation series and compute the square root with error margins. We cover key mathematical concepts such as series, square roots, and error propagation techniques.

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