## Will using Convergent Future give you a critical success if the minimum number you need to hit is 20?

Convergent Future (p185 EGtW) States:

When you or a creature you can see within 60 feet of you makes an attack roll, an ability check, or a saving throw, you can use your reaction to ignore the die roll and decide whether the number rolled is the minimum needed to succeed…

If that number is a “20” does it meet the requirements of a critical success? (p194 PHB)

## Can Convergent Future result in a die roll above 20?

In the new Explorer’s Guide to Wildemount, we are introduced to a new wizard Arcane Tradition called Chronurgy whose capstone ability is:

Convergent Future
14th-level Chronurgy Magic feature

You can peer through possible futures and magically pull one of them into events around you, ensuring a particular outcome. When you or a creature you can see within 60 feet of you makes an attack roll, an ability check, or a saving throw, you can use your reaction to ignore the die roll and decide whether the number rolled is the minimum needed to succeed or one less than that number (your choice).

Say the DC for a Strength ability check is 25, and I have a +2 Strength modifier.
Does Convergent Future allow the die roll to be either 23 (“the minimum needed to succeed”) or 22 (“one less than that number”) even though the maximum possible on a d20 is 20?

## Adding in the Convergent Series

According to the book Introduction to Algorithms, the terms of a convergent series cannot always be added up in any order. I wonder why? Isin’t it just a summation after all?

## Measure of a convergent sequence

Let $$x_k \in R^n, k=1,2,3…$$ be a convergent sequence. Show that the measure of $$\{x_k | k \in N\}$$ is 0. Why is the convergence needed?

## Measure of a convergent sequence

Let $$x_k \in R^n, k=1,2,3…$$ be a convergent sequence. Show that the measure of $$\{x_k | k \in N\}$$ is 0. Why is the convergence needed?

## If $\left\{x_n\right\}$ is a convergent sequence of points in $[a, b]$ and $lim x_n = c$, then $c\in[a, b]$

If $$\left\{x_n\right\}$$ is a convergent sequence of points in $$[a, b]$$ and $$lim x_n = c$$, then $$c\in[a, b]$$.

This is a statement that I found in my real analysis text book. How can I prove the above? Should I use the theorem:

If $$\left\{x_n\right\}$$ and $$\left\{y_n\right\}$$ are two convergent sequences and there exists a natural number $$m$$ such that $$x_n>y_n$$ for all $$n\geq m$$, then $$lim x_n\geq lim y_n$$.

## $\forall a > 0$ $\sum_{n=1}^{\infty} f(na)$ is convergent. Prove that $\int_{0}^{\infty}f(x) dx$ is convergent.
Hi can you help me solve this exercise? Thanks. Let $$f: [0;+\infty) \to \mathbb{R}$$ be nonnegative and continuous function. Suppose $$\forall a > 0$$ $$\sum_{n=1}^{\infty} f(na)$$ is convergent. Prove that $$\int_{0}^{\infty}f(x) dx$$ is convergent. I tried to solve it by using the Riemann sum, but for fixed a it doesn’t work. I have no other ideas.