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4. (a) Express $$ ext{lim}_{eta o 0} extstyle \,\, \\sum_{x=2.1}^{6.3} rac{2}{x} \, ext{d}x $$ as an integral - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 1
Question 6
4. (a) Express $$ ext{lim}_{eta o 0} extstyle \,\, \\sum_{x=2.1}^{6.3} rac{2}{x} \, ext{d}x $$ as an integral. (b) Hence show that $$ ext{lim}_{eta o 0} e... show full transcript
Worked Solution & Example Answer:4. (a) Express $$ ext{lim}_{eta o 0} extstyle \,\, \\sum_{x=2.1}^{6.3} rac{2}{x} \, ext{d}x $$ as an integral - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 1
Step 1
Express lim α→0 ∑ 2αx as an integral
Answer
We start by rewriting the limit as an integral:
ext{lim}_{eta o 0} extstyle \, \\sum_{x=2.1}^{6.3} rac{2}{x} \, ext{d}x = \int_{2.1}^{6.3} \frac{2}{x} \, ext{d}x.This integral expresses the sum as the continuous limit of the Riemann sum as eta approaches zero.
Step 2
Show that lim α→0 ∑ 2αx = ln k
Answer
To find the value of the integral:
Using properties of logarithms, we have:
Thus, we find that:
\text{lim}_{eta o 0} extstyle \, \\sum_{x=2.1}^{6.3} rac{2}{x} \, ext{d}x = \ln(9).This shows that , confirming that:
\text{lim}_{eta o 0} \\sum_{x=2.1}^{6.3} \frac{2}{x} = \text{ln}k,\ ext{ where } k = 9.$